Classification of Transformation groups(変換群の分類)

Shintaro KUROKI(黒木　慎太郎)

Contents

0. Introduction and Acknowledgements 4

Part 1. Classification of compact transformation groups on complex quadrics withcodimension one orbits 7

1. Introduction of Part 1 82. Preliminary 103. Poincare polynomial 114. First step to the classification 255. The two singular orbits are non-orientable 296. One singular orbit is orientable, the other is non-orientable 327. G/K1 ∼ P2n-1(C), G/K2 ∼ S2n 368. G/Ks ∼ Pn(C) 409. P(G/K1; t) = (1 + tk2-1)a(n), k2 is odd: No.1, G/K1 is decomposable. 4310. P(G/K1; t) = (1 + tk2-1)a(n), k2 is odd: No.2, G/K1 is indecomposable. 5111. Compact transformation groups on rational cohomology complex quadrics

with codimension one orbits. 56

Part 2. Equivarinat Graph Cohomology of Hypertorus graph and (n + 1)-dimensional Torus action on 4n-dimensional manifold 59

12. Introduction of Part 2 6013. GKM-graph and equivariant graph cohomology 6214. Hypertoric variety and hypertorus graph 6615. Typical examples 7016. Equivariant graph cohomology of hypertorus graph

—Main theorem and Preparation— 7217. Proof of the main theorem 77

Bibliography 89

3

0. Introduction and Acknowledgements

Let G be a Lie group and X be a smooth manifold. A smooth map ϕ : G × X → X iscalled a (smooth) G-action on X if it satisfies the following two properties:

(1) ϕ(e, x) = x for the identity element e of G and all x ∈ X;

(2) ϕ(g,ϕ(h, x)) = ϕ(gh, x) for all g, h ∈ G and x ∈ X.

We call a triple (G,X, ϕ) a (smooth) transformation group.A transformation group naturally appears in mathematics as the group of automor-

phisms of a manifold with a geometric structure such as a riemaniann metric, for instancethe group of affine transformations on a Euclidean space or the group of rotations ona standard sphere. The first mathematician who recognized the importance of a trans-formation group from the geometrical point of view was Felix Klein. He proposed inhis Erlangen program (in 1872) that geometry is the study of structures invariant undera group action. Since then, the theory of transformation groups has become one of themain research areas in mathematics.

In this thesis we consider the classification problem of transformation groups and itsrelated topics. The first part of the thesis (Part 1) deals with classification of compact Liegroup actions on a rational cohomology complex quadric with codimension one princi-pal orbits. To classify those actions, we use a method developed by Wang [Wan60] andUchida [Uch78]. This method is useful not only to construct interesting examples of com-pact Lie group actions with codimension one principal orbits but also to classify thoseactions.

The second part of this thesis (Part 2) is about equivariant cohomology. The equivariantcohomology H∗

G(X) of a manifold X with G-action is defined to be the ordinary cohomol-ogy of XG := (EG × X)/G where EG is a universal G-bundle and the G-action on EG × X

is the diagonal one. The space XG is called the Borel construction of X. Equivariant coho-mology contains a lot of information about actions and is a useful invariant to distinguishtransformation groups. It is not easy to compute the equivariant cohomology H∗

G(X), butwhen G is a torus T and Hodd(X) = 0, Goresky, Kottwitz and MacPherson [GKM98] de-scribed the image of the restriction map H∗

T (X) → H∗T (X

T ) to the fixed poit set XT undercertain condition. Since the restriction map above is injective because Hodd(X) = 0, theirresult provides a method to compute H∗

T (X). Motivated by this result, Guillemin andZara [GZ01] introduced the notion of GKM-graph (Γ, α, θ) and its equivariant graph coho-mology H∗

T (Γ, α), which is purely combinatorial, in such a way that H∗T (X) is isomorphic to

H∗T (Γ, α) where (Γ, α, θ) is the GKM graph associated with X. In Part 2, we introduce the

notion of a hypertorus graph and its equivariant graph cohomology similarly to Guillemin-Zara’s GKM graph. A hypertorus graph includes a GKM graph which is associated bythe hypertoric or the cotangent bundle of the torus manifold. A hypertorus graph is notnecessarily a Guillemin-Zara’s GKM graph and one can expect to build a new bridgebetween topology and combinatorics as in [GZ01] and [MMP05].

4

My principal gratitude goes to my supervisor Mikiya Masuda, who suggested theproblem of GKM-graphs and helped in numerous ways during the course of the research.Many of the ideas appearing here are outcome of discussion with him. I also would like tothank him to read my paper and correct my English. I learned how to study mathematicsand write a paper from him.

My hearty thanks go on Professor Fuichi Uchida and Megumi Harada. ProfessorUchida is the supervisor of my master course and Harada has done me a great servicewhile I was staying in Toronto. I learned a lot from them and they encouraged me.

I was able to complete this thesis with the help of many excellent teachers: they areTeruko Nagase, Zhi Lu, Taras Panov, Matthias Franz, Akio Kawauchi, Taizo Kanenobu,Yoshitake Hashimoto, Lisa Jeffery, Yael Karshon and many Japanese researchers workingon transformation groups. I am grateful to Yasuzo Nishimura, Shunji Takuma, HiroshiMaeda, Keita Yamasaki, Toshihiro Nogi, Masahiro Kawami, Shigehisa Ishimura, Hiro-masa Moriuchi, Michio Yoshiwaki, Takefumi Miyoshi, Yoshinori Ito, Hisashi Nakayamaand all my academic friends, who explained me some parts of mathematics appearingin the present thesis. I have had countless invaluable conversations with them and allof them have greatly influenced my general mathematical perspective. I feel extremelylucky to be in such a warm and lively community.

The support of my family has been indispensable in completing this thesis. Theyhelped me to become a mathematician.

Finally I gratefully acknowledge the financial support from Japan Society for the Pro-motion of Science.

5

Part 1

Classification of compact transformation groupson complex quadrics with codimension one orbits

1. Introduction of Part 1

One of the central problems in transformation groups is to classify compact Lie groupactions on a fixed smooth manifold M such as a sphere and a complex projective space.Unfortunately the problem is beyond our reach in general, but it becomes within ourreach if we put some assumption on the actions. For instance, when the actions are tran-sitive, M is a homogeneous space and the problem reduces to finding a pair of a compactLie group G and its closed subgroup H such that G/H = M. As is well known, there area rich history and an abundant work in this case (e.g. [BH58], [MS43]). In particular, thetransitive actions on a sphere are completely classified. The complete list can be found in[Aso81] and [HH65].

The orbit of a transitive action is of codimension zero. So we are naturally led to studyactions with codimension one principal orbits. In 1960 H. C. Wang ([Wan60]) initiated thework in this direction. He investigated compact Lie group actions on spheres with codi-mension one principal orbits. In 1977 F. Uchida ([Uch77]) classified compact connectedLie group actions on rational cohomology projective spaces with codimension one prin-cipal orbits. The same problem has been studied by K. Iwata on rational cohomologyquaternion projective spaces ([Iwa78]), on rational cohomology Cayley projective planes([Iwa81]) and by T. Asoh on Z2-cohomology spheres ([Aso81]).

The purpose of Part 1 is to classify compact connected Lie group actions on a ratio-nal cohomology complex quadric with codimension one principal orbits. The complexquadric Qr of complex dimension r is a degree two hypersurface

∑i z2i = 0 in the com-

plex projective space Pr+1(C) of complex dimension r + 1. The linear action of SO(r + 2)on Pr+1(C) leaves Qr invariant and is transitive on Qr. When r is odd, Qr is a rational co-homology complex projective space and this case is already treated by Uchida ([Uch77])mentioned above. Therefore we assume that r = 2n, i.e., our rational cohomology com-plex quadric is of real dimension 4n.

A pair (G, M) denotes a smooth G-action on M and we say that (G, M) is essentiallyisomorphic to (G ′,M ′) if their induced effective actions are isomorphic. Our main theoremis the following.

MAIN THEOREM 1. Let M be a rational cohomology complex quadric of real dimension 4n

and let G be a compact connected Lie group. If (G,M) has codimension one principal orbits, then(G,M) is essentially isomorphic to one of the pairs in the following list.

n G M actionn ≥ 2 SO(2n + 1) Q2n SO(2n + 1) → SO(2n + 2)n ≥ 2 U(n + 1) Q2n U(n + 1) → SO(2n + 2)n ≥ 2 SU(n + 1) Q2n SU(n + 1) → SO(2n + 2)

n = 2m − 1 ≥ 1 Sp(1)× Sp(m) Q4m-2 Sp(1)× Sp(m) → SO(4m)7 Spin(9) Q14 Spin(9) → SO(16)3 G2 Q6 G2 → SO(7) → SO(8)2 S(U(3)×U(1)) Q4 S(U(3)×U(1)) → SO(6)

2 Sp(2) S7 ×Sp(1) P2(C) Sp(2) acts transitively on S7

3 G2 × T 1 GR(2,O)G2 acts naturally and T 1 actsby the induced action from thecanonical SO(2)-action on O2

Here S7×Sp(1) P2(C) denotes the quotient of S7×P2(C) by the diagonal Sp(1)-action where Sp(1)

acts on S7 canonically and on P2(C) through a double covering Sp(1) → SO(3). The manifoldS7 ×Sp(1) P2(C) is not diffeomrophic to Q4 (Proposition 6.2). GR(2,O) denotes a Grassmannmanifold consisting of real 2-planes in the Cayley numbers O. It is diffeomorphic to Q6 (seeSection 7.2).

Closed connected subgroups of SO(r + 2) whose restricted actions on Qr have codi-mension one principal orbits are classified by Kollross [Kol02]. Comparing his result withour list above, we see that the action of G2× T 1 on GR(2,O) ∼= Q6 in the list does not arisethrough a homomorphism to SO(8).

There are some works on compact connected Lie group actions with codimension twoprincipal orbits, see [Nak84] and [Uch77], but the actions get complicated according asthe codimension of principal orbit gets large. The classification of compact connected Liegroup actions with codimension two principal orbits is studied by Uchida ([?]) on rationalcohomology complex projective space. Nakanishi ([Nak84]) completed the classificationof homology spheres with an action of SO(n), SU(n) or Sp(n).

The organization of Part 1 is as follows. In Section 2 we review a key theorem by F.Uchida on compact connected Lie group actions on M with codimension one principalorbits. It says that if H1(M;Z2) = 0, then there are exactly two singular orbits and M

decomposes into a union of closed invariant tubular neighborhoods of the singular orbits.In Section 3 we compute the Poincare polynomials of the singular orbits. To do this,we distinguish three cases according to orientability of singular orbits. In Section 4 wedetermine the possible transformation groups G from the Poincare polynomials using awell known fact on Lie theory([TM]). We also recall some facts used in later sectionsand state an outline of our steps to the classification. Section 5 through 10 are devoted toclassifying the pairs (G, M). By looking at the slice representations of the singular orbits,we completely determine the transformation groups G and the tubular neighborhood ofsingular orbits. Then we check whether the G-manifold obtained by gluing those two

9

tubular neighborhoods along their boundary is a rational cohomology complex quadric.Finally we give all actions in Section 11.

2. Preliminary

In this section, we present some basic facts on a complex quadric and the key theoremto solve the classification problem on a rational cohomology complex quadric. Let usrecall the definition of complex quadric.

Definition(complex quadric Qr).

Qr = {z ∈ Pr+1(C) | z20 + z21 + · · ·+ z2r+1 = 0}

∼= SO(r + 2)/SO(r)× SO(2),

where z = [z0 : z1 : . . . : zr+1] ∈ Pr+1(C).A simply connected closed manifold of dimension 2r is called a rational cohomology

complex quadric if it has the same cohomology ring as Qr with Q coefficient. It is wellknown that the rational cohomology ring of Q2n is given by

H∗(Q2n;Q) = Q[c, x]/(cn+1 − cx, x2, c2n+1),

where deg(x) = 2n, deg(c) = 2.

Let us recall the key theorem about the structure of (G,M).

THEOREM 2.1 (Uchida[Uch77] Lemma 1.2.1). Let G be a compact connected Lie group andM a compact connected manifold without boundary. Assume

H1(M;Z2) = 0,

and G acts smoothly on M with codimension one orbits G(x). Then G(x) ∼= G/K is a principalorbit and (G,M) has just two singular orbits G(x1) ∼= G/K1 and G(x2) ∼= G/K2. Moreover thereexists a closed invariant tubular neighborhood Xs of G(xs) such that

M = X1 ∪ X2 and X1 ∩ X2 = ∂X1 = ∂X2.

Note that Xs is a ks-dimensional disk bundle over G/Ks (ks ≥ 2). The following Figure2.1 is an image of our manifold.

10

G/K1G/K2

G/K

X1 X2

FIGURE 2.1. The image of Theorem 2.1

3. Poincare polynomial

Let M be a rational cohomology complex quadric and G a compact connected Liegroup which acts smoothly on M with codimension one principal orbits. Then the pair(G,M) satisfies the assumptions of Theorem 2.1. Therefore M is devided into X1 and X2where Xi is the tubular neighborhood of singular orbit G/Ki (i = 1, 2). Let us calculatethe Poincare polynomial of the singular orbits G/K1 and G/K2.

First we prepare some notations. Let f∗s : H∗(M;Q) → H∗(Xs;Q) be the homomor-phism induced by the inclusion fs : Xs → M and ns a non-negative integer such thatf∗s(c

ns) 6= 0 and f∗s(cns+1) = 0 where c ∈ H2(M;Q) is a generator. The following theo-

rem is the goal of this section. The result in the case where the two singular orbits areorientable is due to an unpublished note by S. Kikuchi.

THEOREM 3.1. If the two singular orbits are both orientable, then these singular orbits satisfyone of the following.

(1) G/Ks ∼ Pn(C), k1 = 2n = k2, n1 = n = n2.

(2) G/K1 ∼ P2n-1(C), G/K2 ∼ S2n, k1 = 2, k2 = 2n, n1 = 2n − 1, n2 = 0.

(3) P(G/Ks; t) = (1+tkr-1)(1+t2+ · · ·+t2n), k1+k2 = 2n+1, n1 = n = n2, s+r = 3.

If G/K1 is orientable and G/K2 is non-orientable, then• G/K1 ∼ P2n-1(C),• P(G/K2; t) = (1 + t2n), P(G/Ko2 ; t) = (1 + tn)(1 + t2n),• G/Ko ∼ S4n-1,

for n1 = 2n − 1, n2 = 0, k1 = 2, k2 = n.

If the two singular orbits are both non-orientable, then11

• P(G/Ks; t) = 1 + t2 + t4,

• P(G/Kos ; t) = (1 + t2)(1 + t2 + t4),• P(G/K; t) = P(G/Ko; t) = (1 + t3)(1 + t2 + t4),

for n = k1 = k2 = 2 and n1, n2 = 1, 2.

Here M ∼ N means P(M; t) = P(N; t), P(X; t) is the Poincare polynomial of X, and Ko isthe identity component of K.

To prove Theorem 3.1, we will consider three cases according to orientability of twosingular orbits. Before we consider three cases, we shall show the following generalproposition.

PROPOSITION 3.1.(1) n1 + n2 + ε1 + ε2 = 2n.(2) If ε1 = ε2 = 0 then n1 = n2 = n.(3) If ε1 = ε2 = 1 then n1 = n2 = n − 1.

First we show the following three lemmas to prove Proposition 3.1.

LEMMA 3.1. If we put P(Ker f∗s ; t) =∑

tqdim(Ker fqs ) and P(Im f∗s ; t) =∑

tqdim(Im fqs )where Ker fqs = Ker(f∗s) ∩ Hq(M;Q) and Im(fqs ) = Im(f∗s) ∩ Hq(Xs;Q), then the equationP(X3-s, ∂X3-s; t) − tP(Xs; t) = P(Ker f∗s ; t) − tP(Im f∗s ; t) holds.

PROOF. We get dim(Hq(X3-s, ∂X3-s)) = dim(Hq(M, Xs)) by the excision isomorphism.From this equality and the cohomology exact sequence of (M,Xs)

· · · −→ Hq-1(Xs;Q)‹q-1−→ Hq(M,Xs;Q)

jq−→ Hq(M;Q)f∗s−→ Hq(Xs;Q) −→ · · · ,

we get

dim(Hq(X3-s, ∂X3-s)) = dim(Im δq-1) + dim(Ker fqs )

= dim(Hq-1(Xs)) − dim(Im fq-1s ) + dim(Ker fqs ).

Hence we have this lemma. ¤From Lemma 3.1, we can show the following lemma.

LEMMA 3.2. P(Ker f∗1; t) − tP(Im f∗1; t) = t4nP(Im f∗2; t-1) − t4n+1P(Ker f∗2; t

-1).

PROOF. By the Poincare-Lefschetz duality and the universal coefficient theorem weget Hq(Xs) ' H4n-q(Xs, ∂Xs). Hence P(Xs; t) = t4nP(Xs, ∂Xs; t

-1). From Lemma 3.1 we get

P(Ker f∗1; t) − tP(Im f∗1; t) = P(X2, ∂X2; t) − tP(X1; t)

= t4nP(X2; t-1) − t4n+1P(X1, ∂X1; t

-1)

= −t4n+1{P(X1, ∂X1; t-1) − t-1P(X2; t

-1)}

= −t4n+1{P(Ker f∗2; t-1) − t-1P(Im f∗2; t

-1)}.

¤12

Because H∗(M;Q) ' H∗(Q2n;Q), we get the following equations.

LEMMA 3.3. Put εs = 1 if f∗s(x) 6= λf∗s(cn) for all λ ∈ Q, εs = 0 otherwise. Then we have

P(Im f∗s, t) = 1 + t2 + · · ·+ t2ns + εst2n and

P(Ker f∗s, t) = t2ns+2 + · · ·+ t4n + (1 − εs)t2n.

So we can prove Proposition 3.1.

Proof of Proposition 3.1. From Lemma 3.2 and 3.3, we get the following equation

t2n1+2(1 + t2 + · · ·+ t4n-2n1-2) + (1 − ε1)t2n − t(1 + t2 + · · ·+ t2n1) − ε1t

2n+1

= t4n(1 + t-2 + · · ·+ t-2n2) + ε2t2n − t(t4n-2n2-2 + · · ·+ t2 + 1) − (1 − ε2)t

2n+1.

Put t = 1 then we get the first statement in Proposition 3.1. Moreover put ε1 = ε2 = 0 andcompare the degree of this obtained equation by using the first statement then we get thesecond statement. The third statement can be proved similarly.

2

Let us consider three cases according to orientability of two singular orbits.

3.1. Both singular orbits are orientable.Suppose the two singular orbits G/K1 and G/K2 are orientable. First we prove the

following equality.

LEMMA 3.4. Assume ks is the dimension of the normal bundle of G/Ks and s + r = 3, thenthe following equation holds.

(1 − tk1+k2-2)P(G/Ks; t)

= (1 + t-1){P(Im f∗s ; t) + tkr-1P(Im f∗r ; t)} − t-1(1 + tkr-1)P(M; t).

PROOF. By the Thom isomorphism, we get tksP(G/Ks; t) = P(Xs, ∂Xs; t). Since Xs is adeformation retract to G/Ks, P(Xs; t) = P(G/Ks; t). Hence by Lemma 3.1, tkrP(G/Kr; t) −tP(G/Ks; t) = P(Ker f∗s ; t) − tP(Im f∗s ; t). Moreover we get P(G/Kr; t) = tks-1P(G/Ks; t) −t-1P(Ker f∗r ; t)+P(Im f∗r ; t). Using these equations and P(Ker f∗s ; t) = P(M; t)−P(Im f∗s ; t),we can easily check the above equation. ¤

Putting t = −1 in Lemma 3.4, we get (1 − (−1)k1+k2)χ(G/Ks) = (1 − (−1)kr)χ(M)where χ(X) is the Euler characteristic of X. From this equation, we see

LEMMA 3.5. If k1− k2 is even, then k1 and k2 are even. Hence the case k1 ≡ k2 ≡ 1(mod 2)does not occur.

Next we consider two cases.13

3.1.1. The case ε1 = ε2 = 0 and ε1 = ε2 = 1.If ε1 = ε2 = 0 then n1 = n2 = n and if ε1 = ε2 = 1 then n1 = n2 = n−1 by Proposition

3.1. By Lemma 3.3 and 3.4 we have the following equation

fs(t) = (1 + tkr-1)(1 − t2n-1)a(n)(3.1)

where a(n) = 1 + t2 + · · ·+ t2n and fs(t) = (1 − tk1+k2-2)P(G/Ks; t).Suppose k1 ≡ k2 ≡ 0(mod 2). Dividing both sides of the equation (3.1) by 1 + t and

putting t = −1, we get χ(G/Ks) 6= 0 for s = 1, 2. Now we have the following lemma.

LEMMA 3.6. If χ(G/Ks) 6= 0, then the Poincare polynomials P(G/Ks; t) are even functionsfor s = 1, 2 , that is, P(G/Ks; t) = P(G/Ks; −t).

PROOF. If χ(G/Ks) 6= 0, then we have rank Kos = rank G (see [TM] Chapter III). HenceHodd(G/Kos ;Q) = 0 from [TM] Theorem 3.21 in Chapter VII. Sine the induced map

H∗(G/Ks;Q) → H∗(G/Kos ;Q)

is injective, the Poincare polynomials P(G/K1; t) and P(G/K2; t) are even functions. ¤

From this lemma, we see (1 + tkr-1)(1 − t2n-1) = (1 − tkr-1)(1 + t2n-1) by the equation(3.1). Consequently we have k1 = k2 = 2n and G/Ks ∼ Pn(C) because P(Pn(C); t) =1 + t2 + · · ·+ t2n. This means Theorem 3.1 (1).

Suppose k1 is even and k2 is odd. Then both sides of the equation (3.1) are divisibleby 1 − t. Hence we have χ(G/K1) 6= 0. So P(G/K1; t) is an even function. Compare evendegree terms and odd degree terms of the equation (1) then we have k1 + k2 = 2n + 1,P(G/K1; t) = (1 + tk2-1)a(n) and P(G/K2; t) = (1 + tk1-1)a(n). This means Theorem 3.1(3). If k1 is odd and k2 is even, then we get a similar result.

By Lemma 3.5, there does not exist the case that k1 and k2 are odd.

3.1.2. The case ε1 = 0 and ε2 = 1.By Proposition 3.1, Lemma 3.3, Lemma 3.4 and n1 + n2 = 2n − 1, we easily get

f1(t) = (1 − t2n2+k2)a(n1) + (tk2-1 − t2n1+1)a(n2) − t2n-1(1 − tk2),(3.2)f2(t) = (1 − t2n1+k1)a(n2) + (tk1-1 − t2n2+1)a(n1) + t2n(1 − tk1-2).(3.3)

Suppose k1 ≡ k2 ≡ 0(mod 2). Dividing both sides of (3.2), (3.3) by 1 + t and puttingt = −1, we see P(G/K1; t) and P(G/K2; t) are even functions by ks ≥ 2. So k1 = 2n2+ 2 bycomparing the odd degree terms in (3.3). Hence n2 = 0 by comparing the maximal degreeterms in (3.3). So n1 = 2n − 1 and k1 = 2. From (3.2), we see k2 = 2n. ConsequentlyG/K1 ∼ P2n-1(C) and G/K2 ∼ S2n. This result is Theorem 3.1 (2).

Suppose k1 is even and k2 is odd and put t = −1 in (3.2). Then we see P(G/K1; t) is aneven function. So we get

P(G/K1; t) = a(n1) + tk2-1a(n2) + t2n-1+k2 .(3.4)14

Hence dim G/K1 = max {2n1, k2 − 1 + 2n2, 2n − 1 + k2}. If dim G/K1 = 2n1 thenk2 − 1 = 2n1 − (k2 − 1 + 2n2) or 2n1 − (2n − 1 + k2) because of the inequality n ≥ 2, thePoincare duality about G/K1 and the equation (3.4). Hence k2 − 1 = n1 − n2 or n1 − n.Since n1+n2 = 2n−1, n1−n2 is an odd number. Therefore k2−1 = n1−n = n−n2−1 is aneven number. But in this case n = n2 from the Poincare duality. Hence dim G/K1 6= 2n1.If dim G/K1 = k2 − 1 + 2n2, then 2(n2 − n) = k2 − 1 and n2 = n from the Poincareduality. This is in contradiction to k2 ≥ 2. Hence dim G/K1 = 2n − 1 + k2. In this casek2 − 1 ≥ 2n + 2 = 2n1 + 2 from the Poincare duality. So we see dim G/K1 ≥ 4n + 2. Thisis a contradiction. Hence the case k1 is even and k2 is odd does not occur.

Suppose k1 is odd and k2 is even. In this case we get P(G/K2; t) = a(n2)+tk1-1a(n1)+t2n from (3.3). One can easily show that this case does not occur similarly from thePoincare duality.

By Lemma 3.5, there does not exist the case that k1 and k2 are odd.

3.2. Preparation for non-orientable cases.In order to prove two non-orientable cases in Theorem 3.1, it is necessary to show the

following proposition.

PROPOSITION 3.2. If G/K2 is non-orientable, then we have

P(G/Ko2 ; t) = (1 + tk2)P(G/K2; t),

P(G/Ko; t) = (1 + t2k2-1)P(G/K2; t) − P(n1, n2; t) − ε2(1 − ε1)(1 + t-1)t2n,

where

P(n1, n2; t) =

{t2n1+1 + t2n1+2 + · · ·+ t2n2 (n1 < n2)0 (n1 ≥ n2).

The goal of Section 3.2 is to prove Proposition 3.2. Our proof is essentially due toUchida ([Uch77] 2.4, 2.5 and 2.6).

First of all we show the following lemma.

LEMMA 3.7. If k1 > 2, then G/K2 is simply connected, hence K2 is connected.

PROOF. We see π1(M) = π1(G/K2) from k1 > 2, the transversality theorem ([BJ82](14.7)) and Theorem 2.1. Hence G/K2 is simply connected. So K2 = Ko2 because a canonicalmap G/Ko2 → G/K2 is a finite covering. ¤

Next we show the following two lemmas (Lemma 3.8 and 3.9) which just come fromthe condition k1 = 2.

LEMMA 3.8. If k1 = 2, then R∗k = id : H∗(G/Ko;Q) → H∗(G/Ko;Q) for all k ∈ K, whereRk : [g] → [gk] and R∗k is an induced homomorphism of Rk.

15

PROOF. First we assume k2 > 2. Then K1 is connected from Lemma 3.7. BecauseK1/K ∼= S1 by k1 = 2, there is a connected central one dimensional subgroup T in K1 suchthat

K ⊂ K1 = T · Ko.We take a continuous mapping u : [0, 1] → T such that u(0) is the identity in T andu(1) = u ∈ T . Because each u(t) ∈ T commutes with each element in K, a homotopy

Ht : G/Ko → G/Ko

can be defined by Ht(gKo) = gu(t)Ko. For each k ∈ K, there are u ∈ T and k ′ ∈ Ko suchthat k = uk ′. Hence there is u ∈ T such that Rk = Ru for each k ∈ K. Since H0 is theidentity and H1 = Ru = Rk for each k ∈ K, R∗k is the identity map.

Next we assume k2 = 2. By Theorem 2.1, we can put Xs the invariant tubular neigh-borhood of G/Ks (s = 1, 2) in M such that M = X1 ∪ X2 and X1 ∩ X2 = ∂X1 = ∂X2. Letis : X1 ∩ X2 → Xs be the inclusion. Then the induced homomorphism is∗ : π1(X1 ∩ X2) →π1(Xs) is surjective by the transversality theorem ([BJ82] (14.7)). Thus there is a naturalsurjection

hs : π1(Xs) ' π1(X1 ∩ X2)/(Ker is∗) → π1(X1 ∩ X2)/(Ker i1∗) · (Ker i2∗)

such that the following diagram is commutative.

π1(X1 ∩ X2)i1∗−→ π1(X1)

i2∗ ↓ ↓ h1

π1(X2)h2−→ π1(X1 ∩ X2)/(Keri1∗) · (Keri2∗)

Then there is a surjection

π1(X1 ∪ X2) → π1(X1 ∩ X2)/(Ker i1∗) · (Ker i2∗)

by the van Kampen’s theorem. But M = X1∪X2 is simply connected. Hence π1(X1∩X2) =(Ker i1∗)(Ker i2∗). On the other hand, the inclusion is is homotopy equivalent to theprojection ps : G/K → G/Ks. Thus we have

π1(G/K) = (Ker p1∗) · (Ker p2∗).

From homotopy exact seqences for the principal bundles

G → G/K and G → G/Ks,

we have a commutative diagram

π1(G) −→ π1(G/K)„−→ K/Ko

↓ id ↓ ps∗ ↓ ιs

π1(G) −→ π1(G/Ks)„s−→ Ks/Kos

16

where θ and θs are surjective homomorphisms because of π0(G) = {e}. Thus we havefrom π1(G/K) = (Ker p1∗) · (Ker p2∗),

K/Ko = θ(π1(G/K)) = θ((Ker p1∗) · (Ker p2∗))

= θ((Ker p1∗)) · θ((Ker p2∗)) ⊂ (Ker ι1) · (Ker ι2) ⊂ K/Ko

Therefore

K/Ko = ((Ko1 ∩ K)/Ko) · ((Ko2 ∩ K)/Ko) ⊂ (Ko1/Ko) · (Ko2/Ko),

because Ker ιs = (Kos∩K)/Ko. Moreover we see K is a normal subgroup of Ks by Ks/K ∼= S1.Hence there is a connected subgroup T ⊂ Ko1K

o2 of G such that K ⊂ T ·Ko. So we can prove

Lemma 3.8 for k2 = 2 similarly to the case k2 > 2. ¤

From Lemma 3.8, we can show the following lemma.

LEMMA 3.9. If k1 = 2, then H∗(G/Kos ;Q) = Im(q∗s) + Ker(po∗s ) (possibly non direct sum),where q∗s and po∗s are induced from qs : G/Kos → G/Ks and pos : G/Ko → G/Kos .

PROOF. The natural map Kos/Ko → Ks/K is a surjection because Ks/K ∼= Kos/Ko is a(ks − 1)-sphere. So we see Ks = KosK. In particular for each a ∈ Ks there exists k ∈ K suchthat Ra and Rk are homotopic by the connectedness of Kos . Hence R∗a = R∗k : H∗(G/Kos ;Q) →H∗(G/Kos ;Q). By Lemma 3.8 we can consider the following commutative diagram,

H∗(G/Kos ;Q)po∗s−→ H∗(G/Ko;Q)

R∗a = R∗k ↓ R∗k = id ↓H∗(G/Kos ;Q)

po∗s−→ H∗(G/Ko;Q),

for all a ∈ Ks. So we have po∗s (u) = po∗s (R∗a(u)) for u ∈ H∗(G/Kos ;Q) and a ∈ Ks. Ks/Kosacts on H∗(G/Kos ;Q) by R∗k for k ∈ Ks/Kos . Then we easily see Im(q∗s) = H∗(G/Kos ;Q)Ks=K

os .

Hence R∗k(v) = v for all k ∈ Ks/Kos and v ∈ Im(q∗s). Moreover if we put Ks/Kos ={k1, · · · , kr} then R∗k1(u) + · · · + R∗kr(u) ∈ Im(q∗s) for all u ∈ H∗(G/Kos ;Q). Therefore thereis x ∈ H∗(G/Ks;Q) such that po∗s ◦ q∗s(x) = rpo∗s (u). So we see Im(po∗s ) = Im(po∗s ◦ q∗s).Consequently we get the equation H∗(G/Kos ;Q) = Im(q∗s) + Ker(po∗s ). ¤

Put Jk = q∗2Hk(G/K2;Q) and J = ⊕kJk. Next we show properties about this J in the

following two lemmas (Lemma 3.10 and 3.11) by using Lemma 3.9.

LEMMA 3.10. Let χ be the rational Euler class of the oriented (k2−1)-sphere bundle G/Ko →G/Ko2 . If k1 = 2, then Ker(po∗2 ) = J · χ + J · χ2.

PROOF. From the Thom-Gysin exact sequence about po2 : G/Ko → G/Ko2 that is,

po∗2−→ Hq+k2-1(G/Ko2)‹∗−→ Hq(G/Ko2)

·ffl−→ Hq+k2(G/Ko2)po∗2−→ Hq+k2(G/Ko)

‹∗−→,

17

we see Ker(poq2 ) = Hq-k2(G/Ko2 ; Q)·χ. By Lemma 3.9 Hq-k2(G/Ko2 ;Q) = Jq-k2+Ker(poq-k22 ).

So we have Ker(poq2 ) = Jq-k2 ·χ+ Jq-2k2 ·χ2+ · · ·+ Jq-Nk2 ·χN for some integer N. Becauseof the following bundle mapping

G/KoRk−→ G/Ko

↓ po2 ↓ po2

G/Ko2Rk−→ G/Ko2 ,

we see R∗k(χ) = χ or −χ for k ∈ K. Hence R∗k(χ2) = χ2. Since the equation J = Im(q∗2) =

H∗(G/Ko2 ;Q)K2 holds, we have χ2 ∈ J. So we get the equation Ker(po∗2 ) = J · χ + J · χ2. ¤

We remark that non-orientability of G/K2 is not assumed in Lemma 3.7 through 3.10unlike Proposition 3.2. From now on we assume G/K2 is non-orientable. Then k1 = 2

from Lemma 3.7.

LEMMA 3.11. The following two properties hold.(1) dim(Ker(po∗2 )) = dim J + dim(J ∩ Ker(po∗2 )).(2) J · χ ∩ J · χ2 = 0, J · χ2 = J ∩ Ker(po∗2 ) and the homomorphism E : J → Ker(po∗2 ) is

injective, where E is defined by E(y) = y · χ.

PROOF. First we show the property (1) by proving two inequality. From Lemma 3.9we get dim H∗(G/Ko2 ;Q) = dim J + dim(Ker(po∗2 )) − dim(J ∩ Ker(po∗2 )). Since G/K2 isnon-orientable, there is k ∈ K2 such that Rk : G/Ko2 → G/Ko2 reverses an orientation. Sowe see 2dim H∗(G/K2;Q) ≤ dim H∗(G/Ko2 ;Q). Since q∗2 : H∗(G/K2;Q) → H∗(G/Ko2 ;Q) isan injective map, dim J = dim H∗(G/K2;Q). Hence we get

dim J ≤ dim(Ker(po∗2 )) − dim(J ∩ Ker(po∗2 )).

From Lemma 3.10 we get χ2 ∈ J and Jχ2 ⊂ Ker(po∗2 ). So J ·χ2 ⊂ J∩Ker(po∗2 ). Moreoverwe easily see dim(J · χ) ≤ dim J. Hence we get

dim(Ker(po∗2 )) ≤ dim J + dim(J ∩ Ker(po∗2 )).

So we have the property (1) from the two inequalities above .Next we show the property (2). From the equation (1), we have dim(J · χ) = dim J

(so we get the injectivity of E) and dim(J · χ2) = dim(J ∩ Ker(po∗2 )) (so we get J · χ2 =J ∩ Ker(po∗2 )). From Lemma 3.10 Ker(po∗2 ) = J · χ + J · χ2 and J ∩ J · χ = {0}. Hence we getthe property (2). ¤

From Lemma 3.10 and 3.11, we can prove the following equation.

PROPOSITION 3.3. P(G/Ko2 ; t) = (1 + tk2)P(G/K2; t).

PROOF. From Lemma 3.11, we see dim J = dim(Ker(po∗2 )) − dim(J ∩Ker(po∗2 )). More-over from Lemma 3.10 and 3.11 we have the equation

Ker(po∗2 ) = J · χ⊕ J ∩ Ker(po∗2 ).

18

Since χ ∈ Hk2(G/Ko2 ;Q) and dim H∗(G/K2;Q) = dim J, by the equation above we get

P(Ker(po∗2 ); t) = tk2P(G/K2; t) + P(J ∩ Ker(po∗2 ); t).(3.5)

Comparing the equation (3.5) with P(G/Ko2 ; t) = P(Im(q∗2); t) + P(Ker(po∗2 ); t) − P(J ∩Ker(po∗2 ); t), we get P(G/Ko2 ; t) = (1 + tk2)P(G/K2; t) from the injectivity of q∗2. ¤

This result is a part of Proposition 3.2.

Next we show the following equation.

PROPOSITION 3.4. P(G/Ko; t) = (1 + t2k2-1)P(G/K2; t) − (1 + t-1)P(J ∩ Ker(po∗2 ); t).

PROOF. From the Thom-Gysin exact sequence of po2 : G/Ko → G/Ko2 that ispo∗2−→ Hq+k2-1(G/Ko2)

‹∗−→ Hq(G/Ko2)·ffl−→ Hq+k2(G/Ko2)

po∗2−→ Hq+k2(G/Ko)‹∗−→,

we easily get

P(Im(δ∗); t) = P(G/Ko2 ; t) − t-k2P(Ker(po∗2 ); t),(3.6)P(G/Ko; t) = tk2-1P(Im(δ∗); t) + P(Im(po∗2 ); t).(3.7)

From Lemma 3.11 and the injectivity of q∗2,

P(Im(po∗2 ); t) = P(G/K2; t) − P(J ∩ Ker(po∗2 ); t).(3.8)

Substituting (3.7) for (3.6) and (3.8), we obtain the equation

P(G/Ko; t) = tk2-1P(G/Ko2 ; t) − t-1P(Ker(po∗2 ); t)

+ P(G/K2; t) − P(J ∩ Ker(po∗2 ); t).

Moreover substituting the equation above for (3.5) and P(G/Ko2 ; t) = (1 + tk2)P(G/K2; t),the identity of the proposition follows. ¤

Let us concentrate on the term (1 + t-1)P(J ∩ Ker(po∗2 ); t). Consider the followingcommutative diagram

H∗(G/K2;Q)p∗2−→ H∗(G/K;Q)

q∗2 ↓ q∗ ↓H∗(G/Ko2 ;Q)

po∗2−→ H∗(G/Ko;Q),

where q∗ is the induced homomorphism from the natural covering map q : G/Ko → G/K.Now q∗2 is an injection and moreover we show

LEMMA 3.12. q∗ : H∗(G/K;Q) → H∗(G/Ko;Q) is an isomorphism.

PROOF. Let q! : H∗(G/Ko;Q) → H∗(G/K;Q) be the transfer of the covering map q :G/Ko → G/K. From Lemma 3.8 R∗k = id : H∗(G/Ko;Q) → H∗(G/Ko;Q), so q∗ ◦ q! :H∗(G/Ko;Q) → H∗(G/Ko;Q) is r times map where r is the covering degree of q. Hence q∗

is surjective. The injectivity of q∗ is well known. So q∗ is an isomorphism. ¤19

Hence we have Ker(p∗2) = Ker(po∗2 ◦q∗2) ' Im(q∗2)∩Ker(po∗2 ) = J∩Ker(po∗2 ). So we seeP(J ∩ Ker(po∗2 ); t) = P(Ker(p∗2); t). The inclusion is : X1 ∩ X2 → Xs is homotopy equivalentto ps : G/K → G/Ks, hence i∗s = p∗s. Considering the following commutative diagramfrom the cohomology exact sequences of (M,X1) and (X2, X1 ∩ X2) and the excision iso-morphism

H∗(M,X1) −→ H∗(M)f∗1−→ H∗(X1)

'↓ f∗2 ↓ i∗1 ↓H∗(X2, X1 ∩ X2) −→ H∗(X2)

i∗2−→ H∗(X1 ∩ X2),

we get f∗2(Ker(f∗1)) = Ker(i∗2) by this diagram. Hence we obtain the following equationsfrom the definition of n1 and n2, that is f∗s(c

ns) 6= 0 and fs(cns+1) = 0,

P(Ker(i∗2); t) = t2n1+2 + · · ·+ t2n2 + ε2(1 − ε1)t2n (n1 < n2)

and for n1 ≥ n2

P(Ker(i∗2); t) = ε2(1 − ε1)t2n.

Because of the two equations above, P(J ∩ Ker(po∗2 ); t) = P(Ker(i∗2); t) and Proposition ??,we complete the proof of Proposition 3.2.

3.3. G/K1 is orientable, G/K2 is non-orientable.Let us prove where the case one of singular orbits is orientable and the other is not so

in Theorem 3.1. Assume G/K1 is orientable and G/K2 is non-orientable.From Proposition 3.2, we get the following equation.

LEMMA 3.13. t4nP(G/K2; t-1) = t2k2P(G/K2; t).

PROOF. By Proposition 3.2, P(G/Ko2 ; t) = (1+tk2)P(G/K2; t). From the Poincare dualityof G/Ko2 , we see P(G/Ko2 ; t

-1) = tk2-4nP(G/Ko2 ; t). ¤

Since G/K2 is non-orientable, we see k1 = 2 by Lemma 3.7. Hence we can show thefollowing equation.

LEMMA 3.14. P(G/K2; t) = tP(G/K1; t)+a(n2)−t2n2+1a(2n−n2−1)+t2n-1(ε2+tε2−1).

PROOF. Since k1 = 2, we see dim G/K1 = 4n − 2. By the Poincare-Lefschetz dualityand X1 is a deformation retract to G/K1,

Hq(X1, ∂X1;Q) = H4n-q(X1;Q) = H4n-q(G/K1;Q) = Hq-2(G/K1;Q).

So we get the equality P(X1, ∂X1; t) = t2P(G/K1; t).From Lemma 3.1 and 3.3, we have the equation

P(X1, ∂X1; t) − tP(X2; t)

= t2n2+2 + · · ·+ t4n + (1 − ε2)t2n − t(1 + t2 + · · ·+ t2n2 + ε2t

2n)

= t2n2+2a(2n − n2 − 1) − ta(n2) + (1 − ε2 − tε2)t2n.

20

Putting P(X1, ∂X1; t) = t2P(G/K1; t) and P(X2; t) = P(G/K2; t) in this equation, we get thisclaim. ¤

From Lemma 3.13 and 3.14, we can get the following proposition.

PROPOSITION 3.5. P(G/K1; t) is an even function.

PROOF. Multiplying both sides of the identity in Lemma 3.14 by t2k2-1, we get

t2k2-1P(G/K2; t)

= t2k2P(G/K1; t) + t2k2-1a(n2) − t2k2+2n2a(2n − n2 − 1) + t2k2+2n-2(ε2 + tε2 − 1).

Moreover multiplying both sides of the equation which substitute t-1 for t in Lemma 3.13by t4n-1, we get

t4n-1P(G/K2; t-1)

= t4n-2P(G/K1; t-1) + t4n-2n2-1a(n2) − a(2n − n2 − 1) + t2n(ε2 + t-1ε2 − 1).

By the Poincare duality of G/K1, P(G/K1; t) = t4n-2P(G/K1; t-1). From the two equations

above, Lemma 3.13 and the equation P(G/K1; t) = t4n-2P(G/K1; t-1), we get

(1 − t2k2)P(G/K1; t)(3.9)= (1 − ε2)t

2n(1 − t2k2-2) − ε2t2n-1(1 − t2k2)

+ (t2k2-1 − t4n-2n2-1)a(n2) + (1 − t2n2+2k2)a(2n − n2 − 1).

So we easily see χ(G/K1) 6= 0. Hence P(G/K1; t) is an even function. ¤

Since P(G/K1; t) is an even function, it follows from (3.9) that

(t2k2-1 − t4n-2n2-1)a(n2) − ε2t2n-1(1 − t2k2) = 0,(3.10)

(1 − t2k2)P(G/K1; t) = (1 − ε2)t2n(1 − t2k2-2) + (1 − t2n2+2k2)a(2n − n2 − 1).(3.11)

Comparing the minimal degree terms in (3.10), we get k2 = min{2n − n2, n}. If k2 =2n − n2, then we see ε2 = 0 from (3.10). However we see easily χ(G/K1) 6∈ Z from (3.11)and k2 ≥ 2. So this case does not occur.

Hence k2 = n. So we see ε2 = 1 from (3.10).If n2 6= 0, then we see n2 = n−1 from (3.10). In this case we can also prove χ(G/K1) ≡

−(1/n) (mod Z) up to n = 2. Hence χ(G/K1) 6∈ Z. This is a contradiction. Put n = 2, thenwe see G/K1 ∼ P2(C), n2 = 1 and k2 = n = 2. But we see P(G/K2; t) = 1 + t + t2 + t4 − t7

from Lemma 3.14, this contradicts dim Hq(X;Q) ≥ 0.Hence k2 = n, ε2 = 1, n2 = 0. Consequently G/K1 ∼ P2n-1(C) from (3.11). So we get

P(G/K2; t) = 1 + t2n from Lemma 3.14. By Proposition 3.2, P(G/Ko2 ; t) = (1 + tn)(1 + t2n)and G/Ko ∼ S4n-1. This is the case that G/K1 is orientable and G/K2 is non-orientable inTheorem 3.1.

21

3.4. Both singular orbits are non-orientable.Suppose G/K1 and G/K2 are non-orientable. By Lemma 3.7 and Proposition 3.2, we

have k1 = k2 = 2, and

P(G/Kos ; t) = (1 + t2)P(G/Ks; t),(3.12)P(G/Ko; t) = (1 + t3)P(G/Ks; t) − P(nr, ns; t) − εs(1 − εr)(1 + t-1)t2n(3.13)

where

P(n1, n2; t) =

{t2n1+1 + t2n1+2 + · · ·+ t2n2 (n1 < n2)0 (n1 ≥ n2).

3.4.1. The case ε1 = ε2.In this case we see n1 = n2 from Proposition 3.1. So we get the following two equa-

tions from (3.12), (3.13),

P(G/K1; t) = P(G/K2; t),

P(G/Ko; t) = (1 + t3)P(G/Ks; t).

Now we have

P(Im f∗s ; t) = 1 + t2 + · · ·+ t2n

from Lemma 3.3 and Proposition 3.1. We can get the following lemma.

LEMMA 3.15. If M is a rational cohomology complex quadric and P(Im f∗s ; t) = 1+t2+ · · ·+t2n then we have

P(G/K1; t) + P(G/K2; t) = (1 − t2n-1)(1 + t2 + · · ·+ t2n) + P(G/K; t).

PROOF. By the Mayer-Vietoris exact sequence that is

· · · −→ Hq(M)f∗1⊕f∗2−→ Hq(X1)⊕Hq(X2) −→ Hq(X1 ∩ X2) −→ Hq+1(M) −→ · · ·

and the assumptions in the lemma, we see P(X1; t) + P(X2; t) = (1 − t2n-1)(1 + t2 + · · · +t2n) + P(X1 ∩ X2; t). Since Xs is a tubular neighborhood of G/Ks, H∗(Xs) = H∗(G/Ks) andX1 ∩ X2 = G/K. So we get this lemma. ¤

Since ks = 2 (s = 1, 2), we have q∗ : H∗(G/K) → H∗(G/Ko) is an isomorphism. Henceχ(G/K) = χ(G/Ko) = 0. Therefore χ(G/Ks) 6= 0 (that is P(G/Ks; t) is an even functionfrom Lemma 3.6) from P(G/K1; t) = P(G/K2; t) and Lemma 3.15. Substituting Lemma3.15 for P(G/K; t) = P(G/Ko; t) = (1 + t3)P(G/Ks; t) and comparing the degrees, we haven = 2, P(G/Ks; t) = 1 + t2 + t4, P(G/Kos ; t) = (1 + t2)(1 + t2 + t4) and P(G/K; t) =P(G/Ko; t) = (1 + t3)(1 + t2 + t4). This is the case where two singular orbits are bothnon-orientable in Theorem 3.1.

22

3.4.2. The case ε1 6= ε2.In this case we see n1 6= n2 because n1+n2+1 = 2n (Proposition 3.1). We may assume

ε1 = 0 and ε2 = 1. From (3.13), for s = 1,

P(G/Ko; t) = (1 + t3)P(G/K1; t) − P(n2, n1; t),(3.14)

moreover for s = 2

P(G/Ko; t) = (1 + t3)P(G/K2; t) − P(n1, n2; t) − (1 + t-1)t2n.(3.15)

From the Mayer-Vietoris exact sequence, we have the following lemma.

LEMMA 3.16. If M is a rational cohomology complex quadric, then

P(G/K1; t) + P(G/K2; t)

= P(G/K; t) − t-1(1 + t2n)(1 + t2 + · · ·+ t2n) + P(Im f∗1 ⊕ f∗2)(1 + t-1)

From this lemma, we have following two lemmas.

LEMMA 3.17. If n1 < n2, then we have

P(G/K1; t) + P(G/K2; t)

= P(G/K; t) + (1 − t2n+3m-1)(1 + t2 + · · ·+ t2n-3m)

+ t2n-3m+2(1 + t2 + · · ·+ t6m-4) + t2n

where m = χ(G/K1) − χ(G/K2).

PROOF. Suppose n1 < n2. Then we have

(1 + t)(1 − t + t2){P(G/K2; t) − P(G/K1; t)}(3.16)

= t2n1+1(1 + t)(1 + t2 + · · ·+ t2(n2-n1)-2) + (1 + t)t2n-1

from (3.14) and (3.15). From this equation

χ(G/K1) − χ(G/K2) = m = 3-1(n2 − n1 + 1) ∈ Z.(3.17)

Hence n2 − n1 = 3m − 1. Since n2 + n1 = 2n − 1 and n2 > n1, we have

n1 = n −3

2m,

n2 = n − 1 +3

2m

and m(6= 0) is even. Also we have

P(Im(f∗1 ⊕ f∗2); t) = 1 + t2 + · · ·+ t2n2 + t2n

= 1 + t2 + · · ·+ t2n-2+3m + t2n.

Hence we can get the equation in this lemma by Lemma 3.16. ¤23

LEMMA 3.18. If n1 > n2, then we have

P(G/K1; t) + P(G/K2; t)

= P(G/K; t) + (1 − t2n+3m ′+1)(1 + t2 + · · ·+ t2n-3m ′-2)

+ t2n-3m ′(1 + t2 + · · ·+ t6m

′) + t2n

where m ′ = χ(G/K2) − χ(G/K1).

PROOF. Suppose n1 > n2, we get from (3.14), (3.15)

(1 + t)(1 − t + t2)(P(G/K1; t) − P(G/K2; t))(3.18)

= t2n2+1(1 + t)(1 + t2 + · · ·+ t2(n1-n2)-2) − (1 + t)t2n-1.

By this equation

χ(G/K2) − χ(G/K1) = m ′ = 3-1(n1 − n2 − 1) ∈ Z.(3.19)

Consequently n1 − n2 = 1 + 3m ′. So we have

n1 = n +3

2m ′,

n2 = n − 1 −3

2m ′

and m ′ is even, from n1 + n2 = 2n − 1. Also we have

P(Im(f∗1 ⊕ f∗2); t) = 1 + t2 + · · ·+ t2n1 + t2n

= 1 + t2 + · · ·+ t2n+3m ′+ t2n.

Hence we can get the equation in this lemma by Lemma 3.16. ¤

Now we see χ(G/K) = χ(G/Ko) = 0 by Lemma 3.12, (3.14) and (3.15).Hence we have χ(G/K1) + χ(G/K2) = 2n + 2 by Lemma 3.17 and 3.18. Therefore we

can easily show χ(G/Ks) 6= 0 (s = 1, 2) by (3.17) and (3.19). So rank (G) = rank (Kos ) byLemma 3.6. Hence we have Hodd(G/Kos ; Q) = 0. Therefore we see

Hodd(G/Ks;Q) = 0

because of the equation (3.12). Hence if n1 < n2 we have from (3.16),

P(G/K2; t) − P(G/K1; t) = t2n-3m+2a(3m − 2) + t2n

t3(P(G/K2; t) − P(G/K1; t)) = t2n-3m+1a(3m − 2) + t2n-1.

Moreover if n1 > n2 we have from (3.18),

P(G/K1; t) − P(G/K2; t) = t2n-3m ′a(3m ′) − t2n

t3(P(G/K1; t) − P(G/K2; t)) = t2n-3m ′-1a(3m ′) − t2n-1.

By comparing the degrees of these equations, we see the case ε1 6= ε2 does not occur.24

4. First step to the classification

Let G be a compact connected Lie group and U be its maximal rank closed subgroup.The aim of this section is to find the pair (G,U) from Poincare polynomials P(G/U; t)which appeared in Theorem 3.1 up to local isomorphism.

4.1. Equivalence relation.In this section we mention some notations. First we define an essential isomorphism.Definition(essential isomorphism) Put H = ∩x∈MGx. If the induced effective ac-

tions (G/H,M) and (G ′/H ′, M ′) are equivariantly diffeomorphic then we call (G,M) and(G ′,M ′) essential isomorphic.

We will classify (G,M) up to this equivalence relation. Next we define an essentaildirect product.

Definition(essential direct product) Let G1, · · · , Gs be compact Lie groups, and N bea finite normal subgroup of G∗ ' G1 × · · · × Gs. We say that the factor group G = G∗/N

is an essential direct product of G1, · · · , Gs and denote G ' G1 ◦ · · · ◦Gs.

Note that all compact connected Lie groups are constructed by an essential directproduct of some simply connected compact Lie groups and torus (see [TM] Corollary5.31 in Chapter V). Because we would like to classify up to essential isomorphism, we canassume that

G ' G1 × · · · ×Gk × T

for some simply connected simple Lie groups Gi and a torus group T . Moreover we canassume that G acts almost effectively on M where we say that G acts almost effectively onM, if H = ∩x∈MGx is a finite group. In this case G acts almost effectively on the principalorbit G/K, hence we easily see

PROPOSITION 4.1. K dose not contain any positive dimensional closed normal subgroup ofG.

4.2. Candidates for (G,Ks).The purpose of this section is to find the pair (G,U) such that G is a simply connected

compact simple Lie group and U is its maximal rank subgroup where a rank of Lie groupmeans a dimension of a maximal torus subgroup. In Theorem 3.1 we get some evenfunctions P(G/Ki; t). If P(G/Ki; t) is an even function, then rankG = rankKi from Lemma3.6. The following lemma is well known.

LEMMA 4.1 ([TM] Theorem 7.2 in Chapter V). If G ' G1×· · ·×Gk×T then the maximalrank subgroup of G is G ′ ' G ′

1 × · · · ×G ′k × T . Here G ′

i is the maximal rank subgroup of Gi.

Hence we may only find a simply connected compact simple Lie group G and itsmaximal rank closed subgroup U to get (G,Ki) such that P(G/Ki; t) is even. All such

25

pairs (G, U) are known (e.g. [TM], [Wan49]). So we can compute P(G/U; t) by makinguse of the following lemma ([TM] Theorem 3.21 in Chapter VII).

LEMMA 4.2 (Hirsch formula). Let G be a connected compact Lie group and U a max-imal rank connected closed subgroup of G. Suppose H∗(G;Q) ' Λ(x2s1+1, · · · , x2sl+1) andH∗(U;Q) ' Λ(x2r1+1, · · · , x2rl+1) where l = rank G = rank U and xi is an element of thei-th degree cohomology. Then P(G/U; t) satisfies the equation

P(G/U; t) =

l∏

i=1

1 − t2si

1 − t2ri.

From the above argument we get the following propositions. Note that first threepropositions also were known by Uchida [Uch77] Section 4.2.

PROPOSITION 4.2. If P(G/U; t) = 1 + t2a, then (G,U) is locally isomorphic to

(SO(2a + 1), SO(2a)) or (G2, SU(3)), a = 3.

PROPOSITION 4.3. If P(G/U; t) = 1 + t2 + · · · + t2b, then (G,U) is locally isomorphic toone of the following.

(SU(b + 1), S(U(b)×U(1))),

(SO(b + 2), SO(b)× SO(2)), b = 2m + 1,

(Sp(b + 1

2), Sp(

b − 1

2)×U(1)), b = 2m + 1,

(G2, U(2)), b = 5.

26

PROPOSITION 4.4. If P(G/U; t) = (1 + t2a)(1 + t2 + · · · + t2b), then (G,U) is locallyisomorphic to one of the following.

(SO(2m + 2), SO(2m)× SO(2)), a = b = m,

(SO(2m + 3), SO(2m)× SO(2)), a = m,b = 2m + 1,

(SO(7), U(3)), a = b = 3,

(SO(9), U(4)), a = 3, b = 7,

(SU(3), T 2), a = 1, b = 2,

(SO(10), U(5)), a = 3, b = 7,

(SU(5), S(U(2)×U(3))), a = 2, b = 4,

(Sp(3), Sp(1)× Sp(1)×U(1)), a = 2, b = 5,

(Sp(3), U(3)), a = b = 3,

(Sp(4), U(4)), a = 3, b = 7,

(G2, T2), a = 1, b = 5,

(F4, Spin(7) ◦ T 1), a = 4, b = 11,

(F4, Sp(3) ◦ T 1), a = 4, b = 11.

PROPOSITION 4.5. If P(G/U; t) = 1 + t4 + t8 + t12, then (G,U) is locally isomorphic to

(Sp(4), Sp(1)× Sp(3)).

By Theorem 3.1, it is enough to consider above four cases. Before we start the classifi-cation, we outline the proof of the classification.

4.3. Outline of the proof of the classification.In this section we state the outline for the classification. To classify (G,M), where G is

a compact Lie group and M is a rational cohomology complex quadric, we will considerfive cases corresponding to five Poincare polynomials which appeared in Theorem 3.1.Let us recall the following theorem.

THEOREM 4.1 (differentiable slice theorem). Let G be a compact Lie group and M be asmooth G-manifold. Then for all x ∈ M there is a closed tubular neighborhood U of the orbitG(x) ∼= G/Gx and a closed disk Dx, which has an orthogonal Gx-action via the representationσx : Gx → O(Dx), such that G×Gx Dx

∼= U as a G-diffeomorphism.

We call the representation σx in this theorem the slice representation of Gx at x ∈ M.Since we get candidates of singular isotropy groups in Section 4.2, we compute the slicerepresentation of the singular isotropy subgroups K1 and K2 from the differentiable slicetheorem. Then we will decide the transformation group G and two tubular neighbor-hoods X1 ∼= G×K1 Dk1 and X2 ∼= G×K2 Dk2 of two singular orbits G/K1 and G/K2.

Next we construct the G-manifold M up to equivalence by making use of the structuretheorem Theorem 2.1 and the following lemma.

27

LEMMA 4.3 ([Uch77] Lemma 5.3.1). Let f, f ′ : ∂X1 → ∂X2 be G-equivariant diffeomor-phisms. Then M(f) is equivariantly diffeomorphic to M(f ′) as G-manifolds, if one of the followingconditions is satisfied (where M(f) = X1 ∪f X2):

(1) f is G-diffeotopic to f ′.(2) f-1f ′ is extendable to a G-equivariant diffeomorphism on X1.(3) f ′f-1 is extendable to a G-equivariant diffeomorphism on X2.

From Theorem 2.1, we can put ∂Xs = G/K. Hence we may assume the gluing map isin N(K;G)/K, because the set of all G-equivariant diffeomorphisms of G/K is isomorphicto N(K; G)/K where N(K;G) is a normalizer group of K in G.

Finally we compute the cohomology of the manifold which we constructed. And wedecide whether this manifold is a rational cohomology complex quadric or not. This is astory of the classification.

The following two figures are images of classifisation.

X2X1

FIGURE 4.1. Second step of the classification, i.e. compute the slice repre-sentation and find two tubular neighborhoods X1 and X2.

28

X2X1f

G/KFIGURE 4.2. Third step of the classification, i.e. compute the gluing mapf : G/K → G/K.

Let us start to find (G,M) from the next section.

5. The two singular orbits are non-orientable

The goal of this section is to prove this case, that is the two singular orbits are non-orientable, does not occur. By Theorem 3.1, we see P(G/Ks; t) = 1+t2+t4 and P(G/Kos ; t) =(1 + t2)(1 + t2 + t4). So rank G = rank Kos .

5.1. G/Kos is indecomposable.A manifold is called decomposable if it is a product of positive dimensional manifolds.

In this section we consider the case where G/Kos is indecomposable. By Proposition 4.4(a = 1, b = 2), we see G = SU(3)×G ′ × Th and Kos = T 2s ×G ′ × Th. Here T 2s is a maximaltorus of SU(3), G ′ is a product of compact simply connected simple Lie groups and Th isa torus. First we prove the following lemma.

LEMMA 5.1. G = SU(3), Ko1 = Ko2 = T 2 and K1 = K2.

PROOF. Because ks = 2, we see Kos/Ko ∼= S1. Hence G ′ × Th-1 ⊂ Ko from the assump-tion of G ′. Therefore G ′ = {e} and h = 0 or 1 from Proposition 4.1.

To show h = 0, let us consider the slice representation σs : Ks → O(2). Since G/Ks isnon-orientable, there is an element gs ∈ Ks − Kos such that

σs(gs) =

(1 0

0 −1

).

Since the centralizer of σs(gs) in O(2) is a finite group Z2 × Z2 and the centralizer of gs inKs contains {e} × Th, we see {e} × Th ⊂ Ker(σs|Kos ) = Ko where σs|Kos is the ristriction to

29

Kos . Hence h = 0 from Proposition 4.1. Therefore Kos = T 2s which is the maximal torus ofSU(3). Moreover K1 = K2 because K ⊂ K1 ∩ K2 and Ks = KKos . ¤

Next we construct the SU(3)-manifold. To construct the SU(3)-manifold, we will at-tach two tubular neighborhoods along their boundary. So first we consider two tubularneighborhoods of two singular orbits. Put the slice representation σs : Ks → O(2) fors = 1, 2. Since we can assume

T 2 = Kos =

u 0 0

0 v 0

0 0 w

= (u, v, w) ∈ SU(3)

∣∣∣∣∣u, v,w ∈ U(1), uvw = 1

,

the restricted slice representation to T 2 is

σs|T2((u, v,w)) = φ(um)φ(vn)φ(wl)(5.1)

where φ : U(1) → SO(2) is a canonical isomorphism and m,n, l ∈ Z. Now we can easilycheck N(T 2;SU(3))/T 2 is

I =

1 0 0

0 1 0

0 0 1

, A =

0 0 −1

1 0 0

0 −1 0

, A-1 =

0 1 0

0 0 −1

−1 0 0

,

α =

−1 0 0

0 0 1

0 1 0

, β =

0 −1 0

−1 0 0

0 0 −1

, γ =

0 0 1

0 −1 0

1 0 0

.

This group is isomorphic to S3. Hence N(Kos ;SU(3))/Kos ⊃ Ks/Kos ' Z2 or S3 (Kos = T 2) bynon-orientability of SU(3)/Ks. We have following two lemmas.

LEMMA 5.2. If α ∈ Ks, then {(u2, u, u) ∈ SU(3)} ⊂ Ker(σs|Kos ).If β ∈ Ks, then {(u, u, u2) ∈ SU(3)} ⊂ Ker(σs|Kos ).If γ ∈ Ks, then {(u, u2, u) ∈ SU(3)} ⊂ Ker(σs|Kos ).

PROOF. Assume α ∈ Ks. The centralizer of α in Ks contains {(u2, u, u)|u ∈ U(1)}. Thenthe slice representation is σs(u

2, u, u) = σs(α(u2, u, u)α-1) ∈ SO(2). On the other handσs(α(u2, u, u)α-1) = σs(α)σs(u

2, u, u)σs(α)-1 = σs(u2, u, u)-1 because σs(α) ∈ O(2) −

SO(2). This means σs(u2, u, u) = {e} for all u ∈ U(1).

Similarly we can show other cases. ¤LEMMA 5.3. Ks/Kos ' Z2.PROOF. If Ks/Kos ' S3, then Ks = N(Kos ;SU(3)). Hence {α,β, γ,A, A-1} ⊂ Ks. From

Lemma 5.2, {(u2, u, u), (u, u, u2), (u, u2, u)} ⊂ Ker(σs|Kos ). So we see

{(u2, u, u), (u, u, u2), (u, u2, u)} ⊂ Ko.

Hence Ko = T 2 because Ko is a connected Lie subgroup in Kos = T 2. This contradictsKos/Ko ∼= S1. ¤

30

Moreover we can easily see the following lemma from above lemmas and the equation(5.1).

LEMMA 5.4. For m ∈ N, we have the following properties.If {I, α} = Ks/Kos , then Ko = {(u2, u, u)} and σs|Kos (uv, u, v) = φ(um)φ(v-m).If {I, β} = Ks/Kos , then Ko = {(u, u, u2)} and σs|Kos (u, v, uv) = φ(um)φ(v-m).If {I, γ} = Ks/Kos , then Ko = {(u, u2, u)} and σs|Kos (u, uv, v) = φ(um)φ(v-m).

We can easily check Ker(σs|Kos )/Ko ' Zm. Moreover we see σ1|T2 = σ2|T2 . Hence weget the tubular neighborhood

X(m)s = SU(3)×Ks D2

m

where Ks acts on the disk D2m by σs : Ks → O(2) such that Ker(σs|Kos )/Ko ' Zm.

Next we consider an attaching map from X(m)1 to X

(m)2 . Since the attaching map f is

equivariantly diffeomorphic to G/K, f is in N(K;G)/K. Now the following lemma holdsfrom Lemma 5.4 and Ko = T 2.

LEMMA 5.5. N(K; SU(3)) ' U(2).

Hence the attaching map is unique up to equivalence by Lemma 4.3 (1.). So we seesuch an SU(3)-manifold exists for each m ∈ N and

M(m) = SU(3)×Ks S2

where Ks acts on S2 via the linear representation σs : Ks → O(2) such that Ker(σs|Kos )/Ko 'Zm. From above argument, we have the following proposition.

PROPOSITION 5.1. Let M be an SU(3)-manifold which has codimension one orbits SU(3)/K

and two singular orbits SU(3)/Ks (s = 1, 2). Then M is SU(3)-equivariant diffeomorphic toM(m) for some m ∈ N.

Finally we show such an SU(3)-manifold M(m) is not a rational cohomology complexquadric.

PROPOSITION 5.2. M(m) = SU(3)×Ks S2 is not a rational cohomology complex quadric.

PROOF. If M(m) is a rational cohomology complex quadric, then M(m) is simply con-nected. The manifold N = SU(3) ×Kos S2 is a double covering of M(m). Hence M(m) ∼= N.Now N is a fiber bundle over SU(3)/T 2 = SU(3)/Kos with a fiber S2 and SU(3)/T 2 is sim-ply connected. Hence H∗(M(m);Q) ' H∗(N;Q) ' H∗(S2;Q) ⊗ H∗(SU(3)/T 2;Q) becauseHodd(S2;Q) = Hodd(SU(3)/T 2;Q) = 0. Hence H∗(M(m);Q) 6' H∗(Q4;Q). This is a contra-diction. ¤

Hence this case does not occur.31

5.2. G/Ko1 is decomposable.By Proposition 4.2 (a = 1), 4.3 (b = 2), we see that

G = SU(2)× SU(3)×G ′ × Th,

Ko1 = T 1 × S(U(2)×U(1))×G ′ × Th.

First we prove the following lemma.

LEMMA 5.6. G = SU(2)× SU(3) and Ko1 = T 1 × S(U(2)×U(1)) ' Ko2 .

PROOF. If G/Ko2 is indecomposable, then we see Ko2 = SU(2) × T 2 × G ′ × Th. BecauseKo ⊂ Ko1∩Ko2 and Kos/Ko ∼= S1 for s = 1, 2, this is a contradiction. So G/Ko2 is decomposable.Hence we have Ko1 ' Ko2 , G ′ = {e} and h = 0 or 1. Moreover we can show h = 0 likeLemma 5.1. ¤

Because of the non-orientability of

G/Ks,

N(T 1;SU(2))/T 1 ' Z2 andN(S(U(2)×U(1)); SU(3)) = S(U(2)×U(1)),

we have Ks = N(T 1; SU(2))× S(U(2)×U(1)). For the slice representation σs : Ks → O(2),there exists gs ∈ Ks − Kos such that

σs(gs) =

(1 0

0 −1

).

Here the centeralizer of σs(gs) in O(2) is a finite group and the centralizer of gs in Kscontains {e}×S(U(2)×U(1)). Hence S(U(2)×U(1)) ⊂ Ker(σs). So the slice representationσs : Ks → O(2) has a decomposition σs : Ks → N(T 1;SU(2)) → O(2). Moreover Ko ={e}× S(U(2)×U(1)) by Ks/K ∼= S1. Therefore there is an equivariant decomposition

M ∼= (SU(2)×N(T1) D2) ∪@ (SU(2)×N(T1) D2)× P2(C)

where N(T 1) = N(T 1; SU(2)) and as is well known SU(3)/S(U(2)×U(1)) ∼= P2(C). Hencethis case G-manifold is M ∼= N × P2(C) , where N is some SU(2)-manifold (In fact weeasily see N = SU(2) ×N(T1) S2). However this contradicts M is indecomposable. So thiscase does not occur.

6. One singular orbit is orientable, the other is non-orientable

The goal of this section is to prove this case is one of the exotic case in Theorem 1.1.Assume G/K1 is orientable, G/K2 is non-orientable. Then k1 = 2 from Lemma 3.7.

Since k1 = 2, we have K1/K ∼= S1. Let us prove the uniqueness of (G,M).32

6.1. Uniqueness of (G,M).By Theorem 3.1, we see G/Ko ∼ S4n-1, G/K1 ∼ P2n-1(C) (trivially G/K1 is indecompos-

able), P(G/Ko2 ; t) = (1 + tn)(1 + t2n) and P(G/K2; t) = (1 + t2n). Because of K1/K ∼= S1,we get G = H × Th, K1 = H1 × Th (h = 0 or 1) where H is a simply connected simple Liegroup and H1 is its closed subgroup. First we show the following lemma.

LEMMA 6.1. k2 = n = 2 or 4.

PROOF. We see n = k2 from Theorem 3.1. Assume k2 = n is an odd number.Now we have, from Proposition 4.3,

(H,H1) ' (SU(2n), S(U(2n − 1)×U(1))),

(SO(2n + 1), SO(2n − 1)× SO(2)),

(Sp(n), Sp(n − 1)×U(1)) or(G2, U(2)), n = 7.

If (H,H1) = (SU(2n), S(U(2n − 1) × U(1))), then the slice representatoion σ1 : K1→

U(1)'→ SO(2) is as follows;

ρ

((A 0

0 det(A-1)

), x

)= det(A-1)lxm ∈ U(1)

where (l,m) ∈ Z2 − {(0, 0)}. Moreover we see Ker(ρ) = K. Hence we have

Ko ' SU(2n − 1) if h = 0 orKo ' U(2n − 1) if h = 1.

Since k2 = n is an odd number, Ko2/Ko(∼= Sn-1) is an even dimensional sphere. So we seerank Ko2 = rank Ko by χ(Ko2/Ko) 6= 0 and Lemma 3.6. Hence (Ko2 , K

o) is locally isomorphicto one of the following pair

(SO(n), SO(n − 1)),

(G2, SU(3)) if n = 7

from Proposition 4.2. However this contradicts Ko ' SU(2n − 1) or U(2n − 1). Hencewe see k2 = n is an even number for the case (H,H1) = (SU(2n), S(U(2n − 1) × U(1))).Also for other cases we see k2 = n is an even number by the similar argument. Therefroek2 = n is an even number.

Hence we see k2 = n = 2 or 4 from propositions in Section 4.2. ¤

We already have G = H×Th, K1 = H1×Th. Moreover we have Ko2 = H2×Th (h = 0 or1) from Lemma 6.1, where H is a simply connected simple Lie group and Hs is its closed

33

subgroup. By Proposition 4.3, 4.4 and 4.5,

(H,Hos ) ≈ (SU(4), S(U(3)×U(1)) (n = 2),

(Sp(2), Sp(1)×U(1)) (n = 2) or(SO(5), SO(3)× SO(2)) ∼ (Sp(2), U(2)) (n = 2),

(H,H1, Ho2) ≈ (Sp(4), Sp(3)×U(1), Sp(1)× Sp(3)) (n = 4).

where (A1, B1) ≈ (A2, B2) means (A1, B1) and (A2, B2) are locally isomorphic. Since G/K2is non-orientable, N(Ko2 ; G) 6= Ko2 . Hence H = Sp(2) and n = 2. Moreover we see h = 0 bythe similar proof to Lemma 5.1.

Therefore this case has just the following three pairs (G,Ko1 , Ko2).

(G,Kos ) ' (Sp(2), Sp(1)×U(1)),

(G,Kos ) ' (Sp(2), U(2)),

(G,Kos , Kor ) ' (Sp(2), U(2), Sp(1)×U(1))

for s + r = 3. Let us prove the following lemma.

LEMMA 6.2. In this case G = Sp(2), K1 = Sp(1)×U(1), K2 ' Sp(1)×U(1)j ∪U(1)ji andK ' Sp(1)× {1, −1, i, −i} where {1, i, j, k} is the basis of H and U(1)j = {a + bj| a2 + b2 = 1}.

PROOF. Suppose (G,Kos ) ' (Sp(2), U(2)). Since G/K2 is non-orientable, we have K2 'N(U(2); Sp(2)) (K2 has two components). We can put K1 = U(2). So Ko = SU(2) sinceK1/K ∼= S1. Since K1 ∩ K2 ⊃ K and K2 = N(U(2);Sp(2)), we get K2/K ∼= S1 × S1. Thiscontradicts K2/K ∼= S1. So this case does not occur.

Next put (G,Kos , Kor ) ' (Sp(2), U(2), Sp(1) × U(1)) (s + r = 3). Because Kos ⊃ Ko ⊃

SU(2) is not conjugate to Kor ⊃ Ko ⊃ Sp(1) by ks = kr = 2, we have Ko1 ∩ Ko2 = U(2) ∩(Sp(1) × U(1)) = U(1) × U(1) ⊃ Ko. Hence Kos/Ko ∼= S2, this contradicts Ks/K ∼= S1. Sothis case does not occur.

Therefore (G,Kos ) ' (Sp(2), Sp(1) × U(1)). Since G/K1 is orientable and G/K2 is non-orientable, we have K1 = Sp(1) × U(1) = Ko1 and K2 = N(Ko2 ; G). Since Ks/K ∼= S1, wehave K = Sp(1) × F (where F is a finite subgroup of U(1)). If Ko2 = K1 = Sp(1) × U(1),then K2/K ∼= N(U(1); Sp(1))/F ∼= S1 × S1. This contradicts K2/K ∼= S1. So we can putKo2 = Sp(1) × U(1)j without loss of generality. Then K2 = Sp(1) × (U(1)j ∪ U(1)ji) andK1 ∩ K2 = Sp(1)× {1, −1, i, −i}. Since K ⊂ K2 ∩ K1, we have F = {1, −1, i, −i}. ¤

Next we prove the following lemma.

LEMMA 6.3. Let (Sp(2), M) be an Sp(2)-manifold which has codimension one principal or-bits Sp(2)/Sp(1) × {1, −1, i, −i}, two singular orbits Sp(2)/Sp(1) × U(1) and Sp(2)/Sp(1) ×(U(1)j ∪U(1)ji). Then this (Sp(2),M) is unique up to essential isomorphism.

34

PROOF. The slice representations of K1 = Sp(1)×U(1) and K2 = Sp(1)×(U(1)j∪U(1)ji)decompose into the factor as follows:

σ1 : K1 → U(1)1→ O(2),

σ2 : K2 → N(U(1)j; Sp(1)) = U(1)j ∪U(1)ji2→ O(2).

Since Ker(ρ1) = F, we can assume

ρ1(exp(iθ)) =

(cos(4θ) −sin(4θ)sin(4θ) cos(4θ)

)

up to equivalence. So the slice representation σ1 is unique up to equivalence. SinceK2/K ∼= S1 and Ker(ρ2|U(1)j) = {1, −1}, we can put

ρ2(i) = ρ2(−i) =

(1 0

0 −1

).

Therefore the slice representation σ2 is unique up to equivalence. Moreover N(K;G)/K 'U(1)/F has only one connected component. In this case the action is unique by Lemma4.3. ¤

Consequently the following proposition holds.

PROPOSITION 6.1. Let M be an Sp(2)-manifold which satisfies the conditions of Lemma 17.3.Then M ∼= S7 ×Sp(1) P2(C).

PROOF. If M = S7 ×Sp(1) P2(C) where S7 ∼= Sp(2)/Sp(1), Sp(2) acts naturally on S7

and Sp(1) acts on P2(C) = P(R3 ⊗R C) through the double covering Sp(1) → SO(3) (see[Uch77] Example 3.2). Then we can easily check this manifold satisfies the conditions ofLemma 17.3. From Lemma 17.3, we get this proposition. ¤

Hence this case has a unique (G,M) up to essential isomorphism.

6.2. Topology of M = S7 ×Sp(1) P2(C).In this section, we study the topology of M.First we show M is a rational cohomology complex quadric. This manifold M is a

P2(C)-bundle over S7/Sp(1) ∼= S4. Since Hodd(S4) = Hodd(P2(C)) = 0 and S4 is simplyconnected, the induced map p∗ : H∗(S4) → H∗(M) is injective where p : M → S4 is aprojection and i∗ : H∗(M) → H∗(P2(C)) is surjective where i : P2(C) ∼= p-1(w) → M

for fixed w ∈ S4 by [TM] Theorem 4.2 in Chapter III. Hence there exists a generatorx ∈ H4(M) such that x2 = 0 ∈ H8(M) and c ∈ H2(M) such that i∗(c) ∈ H2(P2(C)) isa generator of H∗(P2(C)). Because i∗(x) = 0, we see c2 6= x in H4(M) ' Q ⊕ Q. Nextwe assume S7 × P2(C) a Sp(1)-bundle over M. From the Thom-Gysin exact sequence,H6(M) ' Q is generated by xc and H8(M) ' Q is generated by xc2.

Let us show 0 6= c3 ∈ H6(M). This manifold M has an Sp(2)-action and this ac-tion has codimension one principal orbits from Section 6.1. Therefore we can use the

35

Mayer-Vietoris exact sequence from Theorem 2.1. If we put the principal orbit G/K, theorientable singular orbit G/K1 and the non-orientable singular orbit G/K2, then we haveH∗(G/K) ' H∗(S7) and H∗(G/K2) ' H∗(S4) from Theorem 3.1. Moreover we see, fromSection 6.1, the orientable singular orbit G/K1 is diffeomorphic to P3(C). Hence the in-duced homomorphism j∗ : H2(M) → H2(G/K1) is isomorphic. Therefore j∗(c) is a gen-erator in H2(G/K1) and j∗(c3) = j∗(c)3 6= 0 because H∗(P3(C)) ' Q[c]/(c4). Hence thismanifold M is a rational cohomology complex quadric.

Next we show the tangent bundle of M does not have a spin structure, we call such amanifold non-spin. It is easy to show if a fiber is non-spin then its total space is also non-spin. Hence M is non-spin because P2(C) is non-spin, that is, the second Stiefel-Whitenyclass w2(P2(C)) 6= 0. By definition, Q4 is a degree 2 non-singular algebraic hypersurfacein P5(C). So Q4 is a spin manifold (see Section 16.5 in [BH58] or [MS74]). Therefore M isnot diffeomorphic to Q4.

Hence we get the following proposition.

PROPOSITION 6.2. The 8-dimensional manifold S7 ×Sp(1) P2(C) is not diffeomorphic to Q4,but a rational cohomology complex quadric .

7. G/K1 ∼ P2n-1(C), G/K2 ∼ S2n

The goal of this section is to prove there are three cases (G,M) up to essential isomor-phism. In this case G/K1, G/K2 are indecomposable. Since k1 = 2, k2 = 2n, n ≥ 2 andLemma 3.7, G = H× Th and Ko1 = K1 = H1 × Th (h = 0 or 1). By Proposition 4.3,

(H,H1) ≈ (SU(2n), S(U(2n − 1)×U(1))) or(SO(2n + 1), SO(2n − 1)× SO(2)) or(Sp(n), Sp(n − 1)×U(1)) or(G2, U(2)), n = 3.

Since k1 = 2, we can use Lemma 3.9 and Lemma 3.10. So we have

H∗(G/Ko2 ;Q) = Im(q∗2) + J · χ + J · χ2 (possibly non direct sum)

where q∗2 : H∗(G/K2;Q)(' H∗(S2n;Q)) → H∗(G/Ko2 ;Q) is the injective induced homomor-phism, Jk = q∗2H

k(G/K2;Q) and J = ⊕kJk. Since χ ∈ H2n(G/Ko2 ;Q) by k2 = 2n, we seeJ · χ2 = 0 and J · χ = H2n(G/Ko2 ;Q). Hence P(G/Ko2 ; t) = P(G/K2; t) = 1 + t2n.

Therefore we see (G,Ko2) ≈ (SO(2n + 1), SO(2n)) or (G2, SU(3)) and n = 3 by Propo-sition 4.2. So we have that

(H,H1, H2) = (Spin(2n + 1), Spin(2n − 1) ◦ T 1, Spin(2n)) or(G2, U(2), SU(3)) and n = 3

where Ko2 = H2 × Th.36

7.1. G = Spin(2n + 1)× Th.First we show the following lemma.

LEMMA 7.1. h = 0.

PROOF. If h = 1, then Ko2 = Spin(2n) × T 1. Because G/K2 is orientable, we get K2 =Ko2 . Since k2 = 2n, we have the slice representation σ2 : K2 → SO(2n). From n ≥ 2,we see the restricted representation σ2|Spin(2n) is a natural projection from Spin(2n) onSO(2n). Hence σ2({e}× T 1) ⊂ C(SO(2n)) where C(SO(2n)) is the center of SO(2n) that isC(SO(2n)) = {I2n, −I2n}. Hence {e}× T 1 ⊂ Ker(σ2) ⊂ K. This contradicts Proposition 4.1.So we have h = 0. ¤

Hence we have K1 = Spin(2n − 1) ◦ T 1, K2 = Spin(2n), Ko = Spin(2n − 1). We seeK = Ko from K2/K ∼= S2n-1. Let us prove the following lemma.

LEMMA 7.2. Let (G,M) be a G-manifold which has codimension one orbits G/K, two singularorbits Q2n-1 and S2n where G = Spin(2n + 1), K = Spin(2n − 1). Then this (G,M) is uniqueup to essential isomorphism.

PROOF. Because n ≥ 2, we can decompose the slice representation σ1 : K1 → O(2) intoσ1 : K1 = Spin(2n−1)◦T 1

proj→ T 1→ O(2). Since Ker(σ1) ⊂ K, ρ is an injection. So the slice

representation σ1 is unique up to equivalence. Next we consider the slice representationσ2 : K2 → SO(2n) ⊂ O(2n). Since Z2 ⊂ Ker(σ2) ⊂ σ-1

2 (SO(2n − 1)) = K, σ2 decomposesinto σ2 : K2 = Spin(2n)

proj→ SO(2n)→ SO(2n). Because SO(2n) acts transitively on

S2n-1, we see that ρ is an isomorphism by [HH65] Section I and n ≥ 2. Hence the slicerepresentation σ2 is unique up to equivalence.

Since N(K,G) has two components, we can assume

p(y) =

(−I2n 0

0 1

)

where p : Spin(2n + 1) → SO(2n + 1) is the natural projection, [y] ∈ N(K,G)/N(K,G)o

(y ∈ G = Spin(2n + 1)). It suffices to prove that the right translation Ry on G/K isextendable to a G-diffeomorphic map on X2 from Lemma 4.3 (3.). Because y is in thecenter of K2 = Spin(2n), we have the following commutative diagram

G×K2 K2/K −→ G/K

↓ Ry × 1 ↓ RyG×K2 K2/K −→ G/K.

Here G ×K2 K2/K = ∂(G ×K2 D2m) = ∂X2. It is clear that Ry × 1 is extendable to a G-diffeomorhpic map on X2. ¤

Consequently (G,M) is unique up to essential isomorphism. Such an example of(G,M) will be constructed in Section 11.1. This is one of the results in Theorem 1.1.

37

7.2. G = G2 × Th.The exceptional Lie group G2 is defined by Aut(O). Here O is the Cayley numbers

generated by R-basis {1, e1, · · · , e7}. It is well known that G2 ⊂ SO(7) and SU(3) ' {A ∈G2| A(e1) = e1}.

Let us consider the case h = 0 and 1.

7.2.1. h = 0.In this case K1 ' U(2), Ko2 ' SU(3), Ko ' SU(2). We can put Ko2 = {A ∈ G2| A(e1) = e1}.

Then N(Ko2 , G) has two components. Since G/K2 is orientable and G2/SU(3) ∼= S6, K2 = Ko2and K = Ko. Also in this case (G,M) is unique by the following lemma.

LEMMA 7.3. Let (G2,M) be a G2-manifold which has codimension one orbits G2/SU(2), twosingular orbits G2/U(2) and S6. Then (G2,M) is unique up to essential isomorphism.

PROOF. Because K2 acts transitively on K2/K ∼= S5, the slice representation σ2 is uniqueup to equivalence by [HH65] Section I. Then we see that σ-1

2 (SO(5)) = {B ∈ K2| B(e2) =e2} = K ' SU(2).

The slice representation σ1 decomposes into σ1 : K1 → U(1)→ O(2), because Ker(σ1) ⊂

K. Here ρ is an injection to SO(2). So the slice representation σ1 is unique up to equiva-lence.

Now N(K;G)/K ' SO(3) is known (Section 7.4 in [Uch77]). Consequently (G,M) isunique up to essential isomorphism by Lemma 4.3 (1.). ¤

Hence, in this case, (G,M) is unique up to essential isomorphism. Such an example of(G,M) will be constructed in Section 11.5. This is one of the results in Theorem 1.1.

7.2.2. h = 1.In this case we have G = G2×T 1, K1 = U(2)×T 1, K2 = SU(3)×T 1 and Ko ' SU(2)×T 1,

from the same argument as Section 7.2.1. First we show the following lemma.

LEMMA 7.4. For each natural number m, the pair (G2 × T 1,M(m)), which has codimensionone orbits (G2 × T 1)/K and two singular orbits G2/U(2) and S6, is unique up to equivalence.

PROOF. First we consider the slice representations. Because K2/K ' S5 and σ2({e} ×T 1) ⊂ C(σ2(SU(3)× {e}); SO(6)), where C(X; Y) = {b ∈ Y| ab = ba for all a ∈ X} for X ⊂ Y,the slice representation σ2 : K2 = SU(3)× T 1 → O(6) is as follows

σ2(A + iB, cos(θ) + isin(θ)) =

(A −B

B A

)(cos(mθ)I3 −sin(mθ)I3sin(mθ)I3 cos(mθ)I3

)

for some m ∈ N up to equivalence. Hence

K = σ-12 (SO(5))

=

{((emi„ 0

0 X

), ei„

) ∣∣∣∣∣ det(X) = e-mi„

}.

38

From this equation, we have

K1 = U(2)× T 1

=

{((ei„ 0

0 X

), eiffi

) ∣∣∣∣∣ 0 ≤ θ,φ ≤ 2π, det(X) = e-i„

}.

Moreover we see the slice representation σ1 : K1 = U(2) × T 1→ U(1)

'→ SO(2) is asfollows

ρ

((ei„ 0

0 X

), eiffi

)= ei„e-miffi

because Ker(ρ) = K. Therefore there is a unique pair (σ1, σ2) for each m ∈ N.Next we consider the gluing map. Now we can assume K = SU(2)× T 1 ⊂ SO(7)× T 1

as follows: {((I3 0

0 X

), r

) ∣∣∣∣∣ X ∈ SU(2) ⊂ SO(4), r ∈ T 1

}.

Because N(K; G) = N(K;SO(7)× T 1) ∩ (G2 × T 1), we have

N(K;G)/K ' SO(3).

Consequently (G,M(m)) is unique up to equivalence for each m ∈ N by Lemma 4.3 (1.).Hence we have this lemma. ¤

Next we prepare some notations. Let GR(2,O) be the set of oriented 2-dimensionalreal linear subspace of O. We identify an oriented 2-dimensional real linear subspaces ofO with an element ξ = u ∧ v ∈ Λ2O where u, v ∈ O is an oriented orthonormal basis ofthe 2-dimension subspace. Thus,

GR(2,O) = {ξ ∈ Λ2O| ξ = u ∧ v for some u, v orthonormal in O}

denotes the grassmannian of oriented 2-dimensional suspaces of O. Then this manifoldis diffeomorphic to Q6 (see Section 14 in [Har90]).

Moreover we can show the following proposition.

PROPOSITION 7.1. Let M(m) be a G2 × T 1-manifold which satisfies the conditions of Lemma7.4. Then M(m) ∼= GR(2,O) for all m ∈ N and (G2 × T 1,M(m)) is essential isomorphic to(G2 × T 1,M(1)) for all m ∈ N.

PROOF. Put M = GR(2,O). Assume (G2 × T 1, GR(2,O))(m) is a pair such that g ∈ G2

acts on u∧v ∈ M by g·u∧v = g(u)∧g(v) and ei„ = cos(θ)+isin(θ) ∈ T 1 acts on u∧v ∈ M

by ei„ ·u∧v = (cos(mθ)u−sin(mθ)v)∧(sin(mθ)u+cos(mθ)v). Then we can easily checkthis action is well defined and this pair satisfies the condition of Lemma 7.4. Hence thispair is essentially isomorphic to (G,M(m)). So we can assume M(m) = GR(2,O).

39

Let the action of the pair (G,M(m)) be φ(m). Then Ker φ(m) = {e} × Zm ⊂ G2 × T 1.Hence we see (G, M1) and (G,M(m)) are essentially isomorphic for all m ∈ N. ¤

Hence this case has a unique (G,M) up to essential isomorphism and such action willbe constructed in Section 11.8 again.

8. G/Ks ∼ Pn(C)

In this case Ks = Kos because ks = 2n (n ≥ 2) and Lemma 3.7. First we assume thatG = H1 ×H2 ×G ′ × Th, K1 = H(1) ×H2 ×G ′ × Th, K2 = H1 ×H(2) ×G ′ × Th where Hs is asimply connected simple Lie group, H(s) is its closed subgroup, G ′ is a product of simplyconnected simple Lie groups and Th is a torus. Then K1 ∩ K2 = H(1) ×H(2) × G ′ × Th. Sodim(G/K1 ∩ K2) = 4n ≤ dim(G/K) because K ⊂ K1 ∩ K2. This contradicts dim G/K =4n − 1. Hence we can put

G = H×G ′ × Th,

Ks = H(s) ×G ′ × Th.

where H is a simply connected simple Lie group and H(s) is its closed subgroup. ByProposition 4.3,

(H,H(s)) ≈ (SU(n + 1), S(U(n)×U(1))) or(SO(n + 2), SO(n)× SO(2)), n = 2m + 1 or

(Sp(n + 1

2), Sp(

n − 1

2)), n = 2m + 1 or

(G2, U(2)), n = 5.

Next we show the following lemma.

LEMMA 8.1. If M is a rational cohomology complex quadric, then H = SU(n + 1) andH(s) ' S(U(n)×U(1)).

PROOF. If H(1) acts non-transitively on K1/K ∼= S2n-1, then V = G ′×Th acts transitivelyon K1/K by [MS43] Theorem I’ and K1/K ∼= V/V ′ where V ′ = K ∩ V . So we see p1(K) =H(1) = p1(K1) where p1 : G → H from {pt} = V\K1/K ∼= p1(K1)/p1(K). Hence V\M

is a mapping cylinder of V\G/K1 = H/H(1)∼= V\G/K → V\G/K2 = H/H(2). From the

following commutative diagram

G/K2 −→ M

↓= ↓ p

V\G/K2 = H/H(2)i−→ V\M

40

where i is a homotopy equivalent map, we get the induced diagram

H∗(V\M)i∗−→ H∗(V\G/K2) ' H∗(H/H(2))

↓ p∗ ↓=H∗(M) −→ H∗(G/K2).

From this diagram we see p∗ is an injective map. Put the generator c ∈ H2(V\M) 'H2(H/H(2)). Then p∗(c) = u ∈ H2(M) is a generator. Since cn+1 = 0, we see p∗(c)n+1 =

un+1 = 0. This is a contradiction to un+1 6= 0 from H∗(M) = H∗(Q2n).So H(s) acts transitively on Ks/K ' S2n-1. By making use of [HH65] Section I, we get

(H,H(s)) ' (SU(n + 1), S(U(n)×U(1))). Hence we can put G = SU(n + 1)×G ′ × Th andKs ' S(U(n)×U(1))×G ′ × Th. ¤

Consider the slice representation σs : S(U(n) × U(1)) × G ′ × Th → O(2n). BecauseSU(n) acts transitively on Ks/K ∼= S2n-1, we can assume that σs|SU(n) is a natural inclusionup to equivalence. Hence we can assume σs(Ks) ⊂ U(n) and σs({e} × G ′ × Th) is in thecenter of U(n). This implies G ′ ⊂ Ker(σs) ⊂ K. Hence G ′ = {e} from Proposition 4.1. Sowe can assume the slice representation decomposes into S(U(n)×U(1))× Th

s→ U(n)c→

O(2n) where c is a canonical injective representation. Then we see ρs|S(U(n)×U(1))×feg = τxsfor some integer xs where τxs : S(U(n)×U(1)) → U(n) is

τxs

(A 0

0 det(A-1)

)= (det(A-1))xsA for A ∈ U(n).

Moreover we get K ' (SU(n − 1)× {e}) ◦ Th+1 by Ks/K ∼= S2n-1. From Proposition 4.1, wesee h ≤ 1.

8.1. h = 0.Assume h = 0, then the following lemma holds.

LEMMA 8.2. If h = 0, then G = SU(n + 1), Ks ' S(U(n)×U(1)) and K = S(U(n − 1)×U(1)). Moreover we have xs = 0.

PROOF. Because h = 0, we see G = SU(n + 1), Ks ' S(U(n)×U(1)) and K ' (SU(n −1) × {e}) ◦ T 1. Put the slice representation σs = c ◦ τxs and σ ′s = c ◦ τ-xs where c isa canonical injection c : U(n) → O(2n). Then σs, σ

′s : S(U(n) × U(1)) → O(2n) are

equivalent representations. So Ker(σs) ' Ker(σ ′s). Since the canonical representation c isinjective, Ker(τxs) ' Ker(τ-xs). However if xs 6= 0

Ker(τxs) =

{(A 0

0 det(A-1)

) ∣∣∣∣∣(det(A-1))xsA = In

}

=

{(αIn 0

0 α-n

) ∣∣∣∣∣α ∈ U(1), α-nxs+1 = 1

}.

41

This is a contradiction to Ker(τxs) ' Ker(τ-xs). Hence xs = 0. From τ-10 (U(n − 1)) ' K,

we can put K = S(U(n − 1)×U(1)). ¤

From this lemma, the slice representation σs is unique. Since N(K; G)/K is connected,the attaching map from X1 to X2 is unique up to equivalence by Lemma 4.3 (1.). Hence, inthis case, (SU(n + 1),M) is unique. Such a pair will be constructed in Section 11.2.

8.2. h = 1.Next we put h = 1. In this case the slice representation is

σs : S(U(n)×U(1))× T 1s→ U(n)

c→ O(2n).

Consider the restricted representation of σs to S(U(n) × U(1)). By using the same argu-ment as in Lemma 8.2, we see xs = 0. Hence the representation ρs = ρ

(m)s is

ρ(m)s :

((A 0

0 det(A-1)

), z

)7→ zmA

for some integer m where z ∈ T 1. From Proposition 4.1, we have m 6= 0. Moreover wecan take m > 0 because two slice representations σ

(m)s = c ◦ ρ

(m)s and σ

(-m)s = c ◦ ρ

(-m)s are

equivalent representations.Since (ρ

(m)s )-1(U(n − 1)) = K, we have

K =

z-m 0 0

0 X 0

0 0 zmdet(X-1)

, z

∣∣∣∣∣z ∈ T 1, X ∈ U(n − 1)

Because N(K,G)/K is connected, the attaching map is unique. Hence we get theunique pair (SU(n + 1)× T 1,M(m)) for each m ∈ N, because of Lemma 4.3 (1.). Thereforewe get the following lemma.

LEMMA 8.3. For each natural number m, the pair (SU(n + 1) × T 1,M(m)), which has twosingular orbits (SU(n + 1) × T 1)/Ks and principal orbits (SU(n + 1) × T 1)/K, is unique up toequivalence.

Let us construct such a pair (SU(n + 1)× T 1,M(m)). Take M(m) = Q2n and the SU(n +

1)× T 1-action on Q2n by the representation σ(m) : SU(n + 1)× T 1 → SO(2n + 2) which isdefined by

σ(m) : (A, z) 7→ c(zmA).

Here c : U(n + 1) → SO(2n + 2) is a canonical representation and A ∈ SU(n + 1), z ∈ T 1.We can easily check this pair (SU(n + 1)× T 1, M(m)) has orbits which are the same orbitsin Lemma 8.3. However the following proposition holds.

PROPOSITION 8.1. For all m ∈ N, the pair (SU(n+1)×T 1,M(m)) is essentially isomorphicto (U(n + 1), Q2n) where U(n + 1) acts on Q2n by canonical representation.

42

PROOF. First we put a subgroup

Zn+1 = {(zIn+1, z-1)|z ∈ Zn+1}

which is the center of G = SU(n + 1)× T 1 and the following holds

SU(n + 1)×Zn+1T 1 ' U(n + 1).

Next we consider a kernel of the G-action on M(m) where the kernel of G-action means∩x∈M(m)Gx. Then we have

∩x∈M(m)Gx = Ker(σ(m))

= {(X, z)|zmX = In+1}

= {(z-mIn+1, z)|zm(n+1) = 1}.

Hence Zm ⊂ ∩x∈M(m)Gx. So we see (G, M(m)) is essentially isomorphic to (G,M(1)) for allm ∈ N. Moreover we see Zn+1 = ∩x∈M(1)Gx. Therefore the pair (G,M(1)) is essentiallyisomorphic to (U(n + 1), Q2n). ¤

Hence this case is unique.

9. P(G/K1; t) = (1 + tk2-1)a(n), k2 is odd: No.1, G/K1 is decomposable.

In this case we have K1 = Ko1 because k2 > 2 and Lemma 3.7. Because G/K1 is de-composable, we can put G = H1 × H2 × G" and K1 = H(1) × H(2) × G" where H1/H(1) ∼

Sk2-1, H2/H(2) ∼ Pn(C). Then G/K1 = H1/H(1) ×H2/H(2). So by Propositions 4.2 and 4.3,

(H1, H(1)) = (Spin(k2), Spin(k2 − 1)) or(G2, SU(3)) (k2 = 7).

(H2, H(2)) = (SU(n + 1), S(U(n)×U(1))) or

(Spin(n + 2), Spin(n) ◦ T 1) (n is odd) or

(Sp(n + 1

2), Sp(

n − 1

2)×U(1)) (n is odd) or

(G2, U(2)) (n = 5).

9.1. Preliminary.The goal of this section is to prove the following proposition.

PROPOSITION 9.1. If M is a rational cohomology complex quadric, then H(2) acts transitivelyon K1/K.

In the beginning, we prepare the following lemmas.

LEMMA 9.1 (Theorem I’ in [MS43]). If K × H acts transitively on M, then K or H actstransitively on M.

From Lemma 9.1, we have the following lemma.43

LEMMA 9.2. Let H be a subgroup of G = G1 ×G2 and p : G → G2 be a projection. Then thefollowing two conditions are equivalent.

(1) G1 acts transitively on G/H.(2) p(H) = G2.

Next we prove the following technical lemma.

LEMMA 9.3. Let V ⊂ G be a subgroup such that

π∗ : H∗(V\G/Ks) −→ H∗(V\G/K) is injective,

p∗ : H∗(V\G/Kr) −→ H∗(G/Kr) is injective,

V\G/Kr ∼= V\G/K

where s + r = 3 and π : V\G/K → V\G/Ks and p : G/Kr → V\G/Kr are projections. Thenf∗ : H∗(V\M) → H∗(M) is injective where f : M → V\M is a projection. If M is a rationalcohomology complex quadric, then H2(V\G/Ks;Q) = 0.

PROOF. Assume H2(V\G/Ks;Q) 6= 0. Now V\M is a mapping cylinder of

π : V\G/Kr ∼= V\G/K → V\G/Ks = G/Ks.

Consider a diagram

G/Ksis−→ M

ir←− G/Kr↓ f ↓ p ↓

V\G/Ksjs−→ V\M

jr←− V\G/Kr↓ ↓

V\G/Ksı←− V\G/Kr.

where is, ir, js, jr are natural inclusions. Now js is a homotopy equivalece. This diagraminduces a commutative diagram

H∗(G/Ks)i∗s←− H∗(M)

i∗r−→ H∗(G/Kr)↑ f∗ ↑ p∗ ↑

H∗(V\G/Ks)j∗s←− H∗(V\M)

j∗r−→ H∗(V\G/Kr)↑ ↑

H∗(V\G/Ks)ı∗−→ H∗(V\G/Kr).

From the assumptions, f∗ is an injection.Since we assume H2(V\G/Ks) 6= 0, we can take c ∈ H2(V\M) ' H2(V\G/Ks). Hence

f∗(c2n) = f∗(c)2n 6= 0 because H∗(M) ' H∗(Q2n) where n ≥ 2. Therefore 0 6= c2n ∈H4n(V\G/Ks). This contradicts dim(V\G/Ks) ≤ dim(G/Ks) ≤ dim(M) − 2 = 4n − 2. ¤

Before we prove Proposition 9.1, we show the following lemma.44

LEMMA 9.4. If M is a rational cohomology complex quadric, then H(1)×H(2) acts transitivelyon K1/K.

PROOF. If H(1) × H(2) acts non-transitively on K1/K then G" acts transitively on K1/K

by Lemma 9.1. Hence p(K) = H(1) × H(2) = p(K1) by Lemma 9.2 where p : G → H1 × H2

is the natural projection. Put H1 ×H2 = G ′, H(1) ×H(2) = K ′1 and p(K2) = K ′

2. Then K ′2/K ′

1

is connected, because the induced map p ′ : Sk2-1 ∼= K2/K → K ′2/K ′

1 from p : G → H1 ×H2

is continuous. Hence we see K ′2 is connected from the fibre bundle K ′

1 → K ′2 → K ′

2/K ′1

and the connectedness of K ′1. Now K ′

1 = p(K) ⊂ p(K2) = K ′2 ⊂ G ′. Therefore rank K ′

1 =rank G ′ = rank K ′

2. So we get

P(G/K1; t) = (1 + tk2-1)a(n) = P(G ′/K ′1; t) = P(K ′

2/K ′1; t)P(G ′/K ′

2; t)(9.1)

by the fibration K ′2/K ′

1 → G ′/K ′1 → G ′/K ′

2.Since K2/K ∼= Ko2/Ko is an even dimensional sphere Sk2-1, we see rank Ko2 = rank Ko.

So rank(K1 ∩ Ko2) = rank Ko. Hence Hodd((K1 ∩ Ko2)/K) = Hodd(K ′2/K ′

1) = 0. Because ofthe fibration (K1 ∩Ko2)/Ko → Ko2/Ko

p}→ K ′2/K ′

1 where p" is the induced map from p and thesimply connectedness of Ko2/Ko ∼= Sk2-1 , we see K ′

2/K ′1 is simply connected. Hence we

have

P(Ko2/Ko; t) = 1 + tk2-1 = P(K ′2/K ′

1; t)P((K1 ∩ Ko2)/Ko; t).(9.2)

From equations (9.1) and (9.2), we see H2(G ′/K ′2) = H2(G"\G/K2) 6= 0. Now we

have G"\G/K = G"\G/K1 = G/K1. Moreover we see π∗ : H∗(G"\G/K) = H∗(G ′/K ′1) →

H∗(G ′/K ′2) is injective by the fibration K ′

2/K ′1 → G ′/K ′

1

ı→ G ′/K ′2. This contradicts Lemma

9.3. Therefore H(1) ×H(2) acts transitively on K1/K. ¤

To show Proposition 9.1.1, we prepare some notations.Let pt : G → Ht, p ′t : G → Ht × G" be the natural projection, and let ht : Ht → G,

h ′t : Ht ×G" → G be the natural inclusion. Put

Lst = pt(Ks), Lt = pt(K), L ′st = p ′t(Ks), L ′t = p ′t(K),

Nst = h-1t (Ks), Nt = h-1

t (K), N ′st = (h ′t)

-1(Ks), N ′t = (h ′t)

-1(K).

Then Nst / Lst, Nt / Lt, N ′st / L ′st and N ′

t / L ′t where A / B means a group A is a normalsubgroup of B. In particular L1t = N1t = H(t) and L ′1t = N ′

1t = H(t) × G" by the equalityK1 = H(1) ×H(2) ×G".

Let us prove Proposition 9.1.

Proof of Proposition 9.1. If H(2) does not act transitively on K1/K, then H(1) acts tran-sitively on K1/K by Lemma 9.1 and 9.4. Hence L2 = H(2) = L12 by Lemma 9.2. Then f∗ isan injective homomorphism from Lemma 9.3, where f∗ : H∗((H1 × G")\M) → H∗(M) isan induced homomorphism from the natural projection f : M → (H1 ×G")\M.

Now L22/H(2) is connected because the induced map p ′2 : K2/K → L22/H(2) is continu-ous. Hence L22 is connected by the fibration H(2) → L22 → L22/H(2).

45

Since L2 = H(2) ⊂ L22 ⊂ H2, we have rank H(2) = rank L22 = rank H2 and Hodd(L22/H(2))

= Hodd(H2/L22) = 0. Because L22 is connected, H2/L22 is simply connected. Hence wehave an isomorphism H∗(Pn(C)) ' H∗(H2/H(2)) ' H∗(L22/H(2)) ⊗ H∗(H2/L22) from thefibration L22/H(2) → H2/H(2)

ı→ H2/L22.Assume we can take a ∈ H2m((H1 ×G")\M) ' H2m(H2/L22) 6= 0 for some 0 6= m ≤ n.If m 6= n, then we can put f∗(a) = cm for 0 < m < n where c ∈ H2(M) is a generator.

However there is an l such that n < l + m < 2n and f∗(al) = cl+m 6= 0. This contradictsdim H2/L22 ≤ 2n.

Hence m = n. Then we have H∗((H1 × G")\M) ' H∗(H2/L22) ' H∗(S2n) and we alsohave dim H2/L22 = 2n. Therefore H(2) = L22 from the fibration L22/H(2) → H2/H(2) →H2/L22. Hence we have H2/H(2)

∼= H2/L22. This contradicts H2/H(2) ∼ Pn(C).Therefore H2m(H2/L22) = 0 for m 6= 0. Hence we see L22 = H2. Therfore dim(L22/L2) =

2n. From the fibration (K1 ∩ Ko2)/Ko → Ko2/Ko ∼= Sk2-1 → L22/L2, we see k2 − 1 ≥ 2n. Thiscontradicts k1 + k2 = 2n + 1 and k1 ≥ 2.

2

9.2. Candidates for (G,K1).The goal of this section is to prove k1 = 2n − 2, k2 = 3 and the pair (G,K1) is one of

the following

(G, K1) =

(Sp(1)× Sp(

n + 1

2)×G", T 1 × Sp(

n − 1

2)×U(1)×G"

)

or n = 9,

(G,K1) =(Sp(1)× Spin(11)×G", T 1 × Spin(9) ◦ T 1 ×G"

).

From Proposition 9.1, H(2) acts transitively on K1/K. Then H(2)/N2∼= K1/K ∼= Sk1-1.

Since {pt} = H(2)\K1/K ∼= (H(1) ×G")/L ′1, we have the following lemma.

LEMMA 9.5. L ′1 = H(1) ×G" and L1 = H(1) = L11.

Moreover we can easily show the natural homomorphisms K/(N ′1×N2) → L ′1/N ′

1 andK/(N ′

1 ×N2) → L2/N2 are isomorphic. Hence L ′1/N ′1

∼= L2/N2. Since L2/N2 acts freely onH(2)/N2

∼= Sk1-1, we have the following lemma by [Bre72] 6.2. Theoreom in Chapter IV.

LEMMA 9.6. dim L ′1/N ′1 = dim L2/N2 ≤ 3.

Let us prove the following lemma.

LEMMA 9.7. If M is a rational cohomology complex quadric, then L21 = H1.

PROOF. First we have L21 is connected because K2/K is connected, H(1) = L1 is con-nected and the map p1 : K2/K → L21/L1 = L21/H(1) which induced by p1 : G → H1 iscontinuous. Consider the fibration

L21/H(1) −→ H1/H(1) −→ H1/L21.

46

Then rank H(1) = rank L21 = rank H1 by H(1) = L1 ⊂ L21 ⊂ H1. So we have H∗(H1/H(1)) 'H∗(Sk2-1) ' H∗(H1/L21)⊗H∗(L21/H(1)). Therefore we see L21 = H(1) or H1.

If we put L21 = H(1) = L1, then (H2 × G")\M ∼= [0, 1] ×H1/H(1). Now we consider thefollowing commutative diagram

H1/H(1) ×H2/H(2)∼= G/K1

i1−→ M

↓ q1 ↓ f

H1/H(1)∼= (H2 ×G")\G/K1

j1−→ (H2 ×G")\M.

Here j1 is a homotopy equivalence. Hence q∗1 ◦ j∗1 is injective. Therefore f∗ : H∗((H2 ×G")\M) ' H∗(Sk2-1) → H∗(M) ' H∗(Q2n) is injective. Hence k2 ≥ 2n + 1. But thiscontradicts k1 + k2 = 2n + 1 and k1 ≥ 2. Hence we see L21 = H1. ¤

Hence we can prove the following lemma.

LEMMA 9.8. If M is a rational cohomology complex quadric, then N1 6= H(1).

PROOF. Suppose N1 = H(1), then H(1) ⊂ N21 / L21 = H1 by Lemma 9.7. Since H1 is asimple Lie group, we see N21 = H1. Hence we can put K2 = H1 × X and K = H(1) × X

where X < H2 ×G". Therefore H1\M is a mapping cylinder of H1\G/K = (H2 ×G")/X →H1\G/K1 = H2/H(2). From the following commutative diagram

H1/H(1) ×H2/H(2)∼= G/K1 −→ M

↓ q2 ↓ p

H2/H(2)∼= H1\G/K1

i−→ H1\M

where i is a homotopy equivalent map, we have the following diagram

H∗(H1\M)i∗−→ H∗(H2/H(2))

↓ p∗ ↓ q∗2H∗(M) −→ H∗(H1/H(1))⊗H∗(H2/H(2)).

Hence p∗ is an injection. This contradicts H∗(M) ' H∗(Q2n), H∗(H1\M) ' H∗(Pn(C)). ¤

Next we show the following proposition.

PROPOSITION 9.2. k1 = 2n − 2, k2 = 3 and (H1, H(1)) = (Sp(1), T 1).

PROOF. Let us recall,

(H1, H(1)) = (Spin(k2), Spin(k2 − 1))) or (G2, SU(3)) : k2 = 7.

Because N ′1 ⊂ N1 × G", we have 3 ≥ dim L ′1/N ′

1 ≥ dim H(1) − dim N1 by Lemma 9.5 andLemma 9.6. So we have dim N1 6= 0 if k2 6= 3 because k2 is odd.

If k2 > 6, then H(1) is a simple Lie group. Hence N1 = H(1) from N1 / H(1) = L1 anddim N1 6= 0. This contradicts Lemma 9.8. Hence k2 = 3 or 5.

47

If k2 = 5, then (H1, H(1)) = (Sp(2), Sp(1) × Sp(1)). Then dim N21 ≥ dim N1 > 0

from dim N1 6= 0. Now H1 is a simple Lie group and N21 / L21 = H1 from Lemma 9.7.Hence N21 = H1. This implies K2 = H1 × X where X is a subgroup of H2 × G". BecauseK1 = H(1) × H(2) × G", we see K ⊂ K1 ∩ K2 = H(1) × (X ∩ (H(2) × G")) ⊂ K2. Consider thefibration

(H(1) × (X ∩ (H(2) ×G")))/K → K2/K → K2/(H(1) × (X ∩ (H(2) ×G"))).

Because K2/K ' Sk2-1, K2 = H1×X and dim H1/H(1) = k2−1, we have dim X∩(H(2)×G") =dim X and K = H(1) × Y where dim X/Y = 0. Hence N1 = H(1). This contradicts Lemma9.8. Consequently k2 = 3. Hence k1 = 2n − 2 and (H1, H(1)) = (Sp(1), T 1). ¤

So H(2) acts transitively on S2n-3 from Proposition 9.1 and 9.2. Hence by Proposition4.3 and [HH65] Section I, we have the following two cases where k1 = 2n − 2, k2 = 3,

G = Sp(1)× Sp(n + 1

2)×G",

K1 = T 1 × Sp(n − 1

2)×U(1)×G",

and n = 9,

G = Sp(1)× Spin(11)×G",

K1 = T 1 × Spin(9) ◦ T 1 ×G".

In these cases K2 = Ko2 because n is an odd number and Lemma 3.7 and K = Ko becauseK2/K ∼= S2 is simply connected.

In next two sections we will discuss slice reprepsentations and attaching maps in eachcase.

9.3. G = Sp(1)× Sp(n+12

)×G".If G = Sp(1) × Sp(n+1

2) × G", then K1 = T 1 × Sp(n-1

2) × U(1) × G". Now Sp(n-1

2) ×

U(1) acts transitively on K1/K ∼= S2n-3 because of Proposition 9.1. So we can assumethe restricted slice representation σ1|Sp(n-1

2) is a natural inclusion to SO(2n − 2). Hence

σ1(T1 × {e}×U(1)× G") ⊂ C(σ1(Sp(n-1

2)); SO(2n − 2)) where C(K; G) = {g ∈ G| gk = kg

for all k ∈ K}. We put the natural inclusion σ1|Sp(n-12

) = i : Sp(n-12

) → SO(2n − 2) asfollows:

i(X + Yi + Zj + Wk) =

X −Y Z −W

Y X −W −Z

−Z W X −Y

W Z Y X

.(9.3)

48

Then

C(σ1(Sp(n − 1

2)); SO(2n − 2)) =

h1Im h3Im −h2Im h4Im−h3Im h1Im −h4Im −h2Imh2Im h4Im h1Im −h3Im

−h4Im h2Im h3Im h1Im

(9.4)

where h21 + h22 + h23 + h24 = 1 and m = n-12

. Hence we have

G" ⊂ Sp(1)× Th

where h ≤ 1 by Proposition 4.1 and we can assume the slice representation as σ1 : K1 →Sp(1)× Sp(n-1

2) such that σ1|Sp(n-1

2) : Sp(n-1

2) → {e}× Sp(n-1

2) is isomorphic and σ1(T

1 ×U(1)×G") ⊂ Sp(1)× {e} by (9.3) and (9.4).

Moreover we have the following lemma.

LEMMA 9.9. G" = {e} or T 1 and we can assume the slice representation as σ1 : K1 → U(1)×Sp(n-1

2).

PROOF. Suppose G" = Sp(1)×Th. Then the restricted slice representation σ1|T1×U(1)×G}

is r : T 1×{e}×U(1)×G" → Sp(1). Because Sp(1) is a simple Lie group, r|Sp(1) is isomorphicor trivial representation. If r|Sp(1) is isomorphic, then we have Ker(r) = T 1 × {e}×U(1)×{e} × Th because C(r(Sp(1)); Sp(1)) = {1, −1}. Since Ker(r) ⊂ K, we have H(1) = T 1 ⊂ K.This contradicts the fact H(1) = T 1 6⊂ K from Lemma 9.8. So we see r|Sp(1) is trivial andSp(1) ⊂ Ker(r) ⊂ K. But this contradicts Proposition 4.1. Hence G" = Th for h ≤ 1.Moreover we see easily the slice representation σ1 : K1 → U(1)× Sp(n-1

2). So we get this

lemma. ¤

Assume h = 1. Then we can put the slice representation σ1 : K1 = T 1 × Sp(n-12

) ×U(1)× T 1 → U(1)× Sp(n-1

2) as follows;

σ1

(x,

(A 0

0 y

), z

)7→ (xlymzn, A)

where l,m, n are in Z. Now we can assume the U(1) × Sp(n-12

)-action ρ (via σ1) onS2n-3 ⊂ Hn-1

2 as ρ((t, X), h) = Xht. Hence we have

K =

x,

xlymzn 0 0

0 B 0

0 0 y

, z

∣∣∣∣∣ B ∈ Sp(n − 3

2), x, y, z ∈ T 1

where l 6= 0 by Lemma 9.8.Since K2/K ∼= S2, l 6= 0 and L21 = Sp(1) by Lemma 9.7, we have

K2 =

h,

h 0 0

0 B 0

0 0 y

, z

∣∣∣∣∣ B ∈ Sp(n − 3

2), h ∈ Sp(1), y, z ∈ T 1

.

49

Therefore we have G" = T 1 ⊂ Ker(σ2) ⊂ K by the slice representation σ2 : K2 → SO(3).This contradicts Proposition 4.1. Hence we have G" = {e} that is h = 0.

Moreover, from the same argument, we have

K1 = T 1 × Sp(n − 1

2)×U(1)

K2 =

h,

h 0 0

0 B 0

0 0 y

∣∣∣∣∣ B ∈ Sp(n − 3

2), h ∈ Sp(1), y ∈ T 1

,

K =

x,

x 0 0

0 B 0

0 0 y

∣∣∣∣∣ B ∈ Sp(n − 3

2), x, y ∈ T 1

and

σ1

(x,

(A 0

0 y

))7→ (x,A)

up to equivalence. We also see the slice representation σ2 : K2 → SO(3) is unique up toequivalence.

Next we see

N(K; G)/K ' (N(T 1;Sp(1))/T 1)× (N(U(1); Sp(1))/U(1)).

If we denote by a the generator of N(T 1;Sp(1))/T 1 ' Z2, then xa = ax for all x ∈ T 1.Hence we can consider the following diagram

G×K2 K2/Kf−→ G/K

↓ 1× R¸ ↓ R¸

G×K2 K2/Kf−→ G/K.

Here f([g, kK]) = gkK and α = (a, e, e) ∈ N(K; K2). We have gkKα = gkαK for all g ∈ G

and k ∈ K2. So this diagram is commutative. In this case R¸ is the antipodal involutionon K2/K ∼= S2. Hence R¸ is extendable to a K2-equivariant diffeomorphism on D3. HenceM(R¸) ∼= M(id) from Lemma 4.3 (3.). Since N(U(1); Sp(1))/U(1) ' Z2, there are just twomanifolds up to essential isomorphism. Hence we get the following proposition.

PROPOSITION 9.3. Let (G,M) be a G-manifold which has codimension one orbit G/K andtwo singular orbit G/K1 and G/K2 where G = Sp(1)× Sp(n+1

2), K1 = T 1 × Sp(n-1

2)×U(1),

K2 =

h,

h 0 0

0 B 0

0 0 y

∣∣∣∣∣ B ∈ Sp(n − 3

2), h ∈ Sp(1), y ∈ T 1

and

K =

x,

x 0 0

0 B 0

0 0 y

∣∣∣∣∣ B ∈ Sp(n − 3

2), x, y ∈ T 1

.

50

Then there are just two such (G,M) up to essential isomorphism which are M = Q2n and M =

(Sp(1)× Sp(k + 1))×Sp(1)×Sp(k)×U(1) S4k+2 where k = n-12

.

PROOF. By the above argument, this case has just two types up to essential isomor-phism. If M = Q2n, then this case will be realized in Section 11.3. If M = (Sp(1)× Sp(k +

1))×Sp(1)×Sp(k)×U(1) S4k+2 such that k = n-1

2and S4k+2 ⊂ R3×Hk has the trivial U(1)-action,

the canonical Sp(1)-action on R3 and the canonical Sp(1)× Sp(k)-action on Hk. Then thismanifold has the Sp(1)× Sp(k + 1)-action. We can easily check this manifold satisfies theassumption of this proposition. ¤

M = (Sp(1)×Sp(k+1))×Sp(1)×Sp(k)×U(1) S4k+2 is the fibre bundle over Sp(k+1)/U(1)×

Sp(k) ∼= P2k+1(C) with the fibre S4k+2. We see easily check Hodd(P2k+1(C)) = Hodd(S4k+2) =0 and P2k+1(C) is simply connected. Hence p∗ : H∗(P2k+1(C)) → H∗(M) is injective wherep : M → P2k+1(C) is a projection. Hence the 2k + 2 times cup product of c ∈ H2(M) isvanishing in H4k+4(M). Hence this is not a rational cohomology complex quadric. So thiscase is unique up to essential isomorphism and such (G,M) will be constructed in Section11.3.

9.4. G = Sp(1)× Spin(11)×G".If G = Sp(1)×Spin(11)×G", then we have K1 = T 1×Spin(9)◦T 1×G" and G" = {e} or T 1.

Put the slice representation σ1 : K1 → O(16). Then the restricted representation σ1|Spin(9)

is the spin representation to SO(16) and we can easily show C(σ1(Spin(9)); SO(16)) is afinite group. So we have σ1(T

1 × {e}) = {I16} because T 1 × {e} ⊂ C(Spin(9); K1), wheree ∈ Spin(9) ◦ T 1 × G" and I16 ∈ O(16) are identity elements. Therefore we see K ⊃Ker(σ1) ⊃ T 1 × {e}. So N1 = h-1

1 (K) = T 1 = H(1), recall h1 denotes the natural inclusionH1 → G. This contradicts Lemma 9.8. Hece this case does not occur.

10. P(G/K1; t) = (1 + tk2-1)a(n), k2 is odd: No.2, G/K1 is indecomposable.

In this case K1 = Ko1 by k2 > 2 and Lemma 3.7. Because G/K1 is indecomposable, wecan put G = G ′ × G" and K1 = K ′

1 × G" where G ′ is a simple Lie group and G" is a directproduct of some simple Lie groups and a toral group. By Proposition 4.4, (G ′, K ′

1)-pairwhich satisfies

P(G/K1; t) = P(G ′/K ′1; t) = (1 + t2a)(1 + t2 + · · ·+ t2b)

51

where 2a = k2 − 1 and b = n is one of the following thirteen pairs

(SO(2n + 2), SO(2n)× SO(2)), a = b = n,

(SO(k2 + 2), SO(k2 − 1)× SO(2)), a = (k2 − 1)/2, b = k2,

(SO(7), U(3)), a = b = 3,

(SO(9), U(4)), a = 3, b = 7,

(SU(3), T 2), a = 1, b = 2,

(SO(10), U(5)), a = 3, b = 7,

(SU(5), S(U(2)×U(3))), a = 2, b = 4,

(Sp(3), Sp(1)× Sp(1)×U(1)), a = 2, b = 5,

(Sp(3), U(3)), a = b = 3,

(Sp(4), U(4)), a = 3, b = 7,

(G2, T2), a = 1, b = 5,

(F4, Spin(7) ◦ T 1), a = 4, b = 11,

(F4, Sp(3) ◦ T 1), a = 4, b = 11.

In the beginning, we will find the candidates for (G ′, K ′1).

10.1. Candidates for (G ′, K ′1).

The goal of this section is to prove the pair (G ′, K ′1) is one of the following

(Spin(9), Spin(6) ◦ T 1) (k1 = 8, k2 = n = 7) or(SU(3), T 2) (k1 = 2, k2 = 3, n = 2).

Now k1 ≥ 2 and k1 + k2 = 2n + 1. So we can easily see the following three cases donot satisfy k1 ≥ 2. (Recall a = k2-1

2and b = n.)

(SO(2n + 2), SO(2n)× SO(2)), a = b = n,

(SO(7), U(3)), a = b = 3,

(Sp(3), U(3)), a = b = 3.

We can show the following proposition similarly to Proposition 9.1.

PROPOSITION 10.1. K ′1 acts transitively on K1/K ∼= Sk1-1.

52

Hence we see the following six cases contradict Proposition 10.1 by the paper [HH65]Section I.

(SO(k2 + 2), SO(k2 − 1)× SO(2)), a = (k2 − 1)/2, b = k2,

(SO(10), U(5)), a = 3, b = 7,

(Sp(3), Sp(1)× Sp(1)×U(1)), a = 2, b = 5,

(G2, T2), a = 1, b = 5,

(F4, Spin(7) ◦ T 1), a = 4, b = 11,

(F4, Sp(3) ◦ T 1), a = 4, b = 11.

Therefore in this case we have that

(G ′, K ′1) = (Spin(9), Spin(6) ◦ T 1) (k1 = 8, k2 = n = 7) or

(SU(3), T 2) (k1 = 2, k2 = 3, n = 2) or(SU(5), S(U(3)×U(2))) (k1 = 4, k2 = 5, n = 4) or(Sp(4), U(4)) (k1 = 8, k2 = n = 7).

If (G ′, K ′1) = (SU(5), S(U(3)×U(2))), then k1 = 4. Hence K1/K ∼= S3. Since U(2) (⊂ K ′

1)acts transitively on K1/K by Proposition 10.1, we can assume the slice reprepsentationas σ1 : K1 → U(2). Therefore we see G" = Th (h ≤ 1) and K ' S(U(3) × {e}) ◦ T 1+h

by Proposition 4.1 and Proposition 10.1. In particular we see K2 ⊃ K ⊃ SU(3). SinceK2/K ∼= S4, (K2, K) = (A ◦ N,B ◦ N) where (A,B) ≈ (SO(5), SO(4)) by Proposition 4.2.Now we easily see N ⊃ SU(3). So K2 ⊃ A ◦SU(3). That is dim K2 ≥ dim(A ◦SU(3)) = 18.However we have dim(K2) = 13 or 14 by dim(K) = dim(S(U(3) × {e}) ◦ T 1+h) = 9 + h

(h ≤ 1) and K2/K ∼= S4. This is a contradiction. Hence this case does not occur.

If (G ′, K ′1) = (Sp(4), U(4)), then k1 = 8 and K1/K ∼= S7. From Proposition 10.1, we can

assume the slice reprepsentation as σ1 : K1 → U(4). So G" = {e} or T 1 by Proposition 4.1.Since K2/K ∼= S6 and K1 = U(4) or U(4) × T 1, we have (K2, K) ≈ (G2 ◦ T 1, SU(3) ◦ T 1)or (G2 ◦ T 2, SU(3) ◦ T 2) by Proposition 4.2. Therefore we get Sp(4) ⊃ G2. However thefollowing proposition holds.

PROPOSITION 10.2. Sp(4) 6⊃ G2.

PROOF. Assume Sp(4) ⊃ G2. Let V be the Sp(4)-C irreducible 8-dimensional repre-sentation space (complex dimensional). Then we can consider Sp(4) acts effectively onV by the natural representation ρ : Sp(4) → U(8). Since Sp(4) ⊃ G2 and Ker(ρ) = {e},we see the restricted reprepsentation to G2 ρ|G2 is not trivial. Because the least dimensionof non-trivial complex representation of G2 is 7 and V is an 8-dimensional space, there isan irreducible decomposition V = V7 ⊕W where V7 is a complex seven dimensional G2-space which has a representation ρ|G2 and W is a complex one dimensional space whichhas trivial G2-action. Then V has the structure map J : V → V such that J is a Sp(4) map,

53

J2(v) = −v and J(zv) = zJ(v) for z ∈ C and v ∈ V (see [Ada69] 3.2). Moreover J(w) ∈ W

for w ∈ W because J is a G2(⊂ Sp(4)) map. However W is a complex one dimensionalspace, so this contradicts W does not have such map. Therefore we see Sp(4) 6⊃ G2. ¤

Hence the following two cases remain.

10.2. (G ′, K ′1) = (Spin(9), Spin(6) ◦ T 1).

If (G ′, K ′1) = (Spin(9), Spin(6) ◦ T 1), then k1 = 8. So K1/K ∼= S7, hence G" = Th (h ≤ 1)

from Proposition 4.1 and Proposition 10.1.Assume h = 1. Since K2/K ∼= S6, we see (K2, K) = (G2 ◦ T 2, SU(3) ◦ T 2). Consider

the slice representation σ2 : G2 ◦ T 2 → SO(7). Because K2 acts transitively on K2/K ∼= S6,the restricted representation σ2|G2 is a natural inclusion. So C(σ2(G2); SO(7)) = {e} whereC(K;G) = {g ∈ G| gk = kg for all k ∈ K}. Therefore G" ⊂ Ker(σ2) = T 2 ⊂ K. Now G" = T 1

is a normal subgroup of G. This contradicts Proposition 4.1. Hence h = 0.We get G" = {e} and (G,K1) = (Spin(9), Spin(6) ◦ T 1). Since h = 0 and K2/K ∼= S6, we

see (K2, K) = (G2 ◦ T 1, SU(3) ◦ T 1). Hence we can easily show that slice representationsσ1 : K1 → SO(8) and σ2 : K2 → SO(7) are unique up to equivalence (σ1 through Spin(6) 'SU(4)). Moreover we see N(K;G)/N(K;G)o = Z2. Hence in this case there are just twoG-manifolds M up to essential isomorphism. Hence the following proposition holds.

PROPOSITION 10.3. Let (Spin(9),M) be a Spin(9)-manifold which has codimension oneorbits Spin(9)/SU(3) ◦ T 1 and two singular orbits Spin(9)/K1 and Spin(9)/K2 where K1 =Spin(6) ◦ T 1 and K2 = G2 ◦ T 1. Then there are just two such (Spin(9),M) up to essentialisomorphism, that is, M = Q14 and M = Spin(9)×Spin(7)◦T1 S14.

PROOF. From the above argument this case has just two such (Spin(9),M) up to es-sential isomorphism. If M = Q14, then we will be constructed in Section 11.4. PutM = Spin(9) ×Spin(7)◦T1 S14 such that T 1 acts S14 ⊂ R8 × R7 trivially and Spin(7) actscanonically on R7 and acts on R8 through the spin representation Spin(7) → SO(8). Thenthis manifold has a canonical Spin(9) action and satisfies the assumption of this case. ¤

But M = Spin(9)×Spin(7)◦T1 S14 is the fibre bundle over Spin(9)/Spin(7) ◦ T 1 ∼= P14(C)

with the fibre S14. Hence this is not a rational cohomology complex quadric. So this case isunique up to essential isomorphism and such (G,M) will be constructed in Section 11.4.

10.3. (G ′, K ′1) = (SU(3), T 2).

If (G ′, K ′1) = (SU(3), T 2), then k1 = 2. Hence G" = Th and h ≤ 1. From K2/K ∼= S2 and

Proposition 4.2, we have Ko2 ' SU(2) ◦N and Ko ' T 1 ◦N.If h = 0 then we have N = {e} because K1/Ko ∼= S1. Then the slice reprepsentation

σ1 : K1 = T 2 → U(1)(→ O(2)) is

σ1(x, y) = xmyn

54

where m, n ∈ Z and (m,n) 6= 0. We see Ker(σ1) = K and we can put T 2 = {(x-1y-1, x, y) ∈SU(3)} and m ≥ n without loss of generality where (x, y, z) is a diagonal matrix in SU(3),then

K = {(x-1y-1, x, y)| x, y ∈ T 1 and xmyn = 1}.

Hence we have K ' T 1 × F where F is a finite group and then K2 ' SU(2) × F. Moreoverwe see the slice representation σ2 : K2 → SO(3) is unique up to equivalence because therestricted representation σ2|SU(2) is a canonical double covering and Z2 ◦ F = Ker(σ2).

Next we discuss N(K; G)/N(K; G)o. First of all we define the following notations.

α =

−1 0 0

0 0 1

0 1 0

, β =

0 −1 0

−1 0 0

0 0 −1

, γ =

0 0 1

0 −1 0

1 0 0

.

We can easily show the next four statements.(1) If m = n( 6= 0), we have N(K;G)/N(K;G)o = {I3, α}.(2) If m = 0 > n, we have N(K;G)/N(K;G)o = {I3, β}.(3) If m > n = 0, we have N(K;G)/N(K;G)o = {I3, γ}.(4) If m > n and mn 6= 0, we have N(K; G)/N(K; G)o = {I3}.

We also see Ko is conjugate to the following subgroup of G except the last case above.

1 0 0

0 x 0

0 0 x-1

∣∣∣∣∣ x ∈ T 1

.

Hence P(G/Ko; t) = (1 + t2)(1 + t5) from the fibration SU(2)/Ko ∼= S2 → G/Ko →G/SU(2) ∼= S5. Therefore we have, from the fibration Ko2/Ko ∼= S2 → G/Ko → G/Ko2 ,the Poincare polynomial of G/Ko2 is P(G/Ko2 ; t) = 1 + t5. This contradicts P(G/K2; t) =(1 + t)(1 + t2 + t4) and an injectivity of p∗ : H∗(G/K2;Q) → H∗(G/Ko2 ;Q).

Therefore N(K; G)/N(K; G)o = {I3}. Hence N(K;G)/K is connected and the attachingmap is unique up to equivalence by Lemma 4.3 (1.). So we can put such SU(3)-manifold asM = SU(3)×S(U(2)×U(1)) S

4 where S(U(2)×U(1)) acts on S4 ⊂ R3×R2 by S(U(2)×U(1))p1→

SU(2)c→ SO(3) and S(U(2) × U(1)

p2→ T 1fiF→ SO(2) where p1, p2 are projections, c is

a canonical double covering and Ker(τF) = F. The manifold M is a fibre bundle overSU(3)/S(U(2) × U(1)) ∼= P4(C) with fibre S4. This is not a rational cohomology complexquadric. Hence this case does not occur.

Hence h = 1, G = SU(3) × T 1 and K1 = T 2 × T 1. Moreover we see N = T 1 becauseK1/K ∼= S1. Hence K2 ' SU(2) ◦ T 1 × F and K ' T 2 × F where F ⊂ T 1 is a finite subgroup.Then we can show easily the slice representation σ1 decomposes into K1 → T 1

→ O(2)

such that Ker(ρ) = F. Moreover σ2 decomposes into σ2 : K2 → SU(2)fi→ SO(3) where τ is

a canonical double covering. Hence σ2 is unique up to equivalence.55

Because N(K; G)/N(K; G)o = N(K1;G)/K1 = Z2 × Z3 and Z3 ⊂ K2, this case has justtwo (G,M) up to essential isomorphism for the fixed positive integer m = |F|. Hence thefollowing proposition holds.

PROPOSITION 10.4. Let (G,M) be a G-manifold which has codimension one orbit G/K, andtwo singular orbits G/K1 and G/K2 where G ' SU(3) × U(1). Then this (G,M) has just twotypes up to essential isomorphism, that is, M = Q4 and M = SU(3)×S(U(2)×U(1)) S4.

PROOF. There are two types (G,M) from above argument for the positive integer m.Put M = Q4, and the representation rm : SU(3)× T 1 → S(U(3)×U(1)) such that

rm(A, x) =

(x-m=6A 0

0 xm=2

).

As in Section 11.6, there is a representation ρ : S(U(3) × U(1)) → SO(6) from the naturaldouble covering surjection SU(4) → SO(6). Then Ker(ρ ◦ rm) = {I3} × F. Hence theSU(3) × T 1 acts on Q4 by ρ ◦ rm such that K = T 2 × F. Moreover we see, for all m = |F|,the induced effective actions are equivariantly diffeomorphic. Therefore such action isunique up to essential isomorphism.

Put M = SU(3)×S(U(2)×U(1)) S4 such that S(U(2)×U(1)) acts on S4 ⊂ R3 ×R2 throughthe representation S(U(2) × U(1)) → SO(3). This manifold has the action of SU(3) × T 1,that is, SU(3) acts on SU(3) canonically and T 1 acts on S4 ∩ R2 by m-fold. Then thisSU(3)× T 1-manifold M satisfies the assumption of this case. We can assume T 1-action onS4 ∩ R2 is canonical because all m-fold actions are essentially isomorphic. ¤

The manifold M = SU(3)×S(U(2)×U(1)) S4 is an S4-bundle over P4(C) ∼= SU(3)/S(U(2)×

U(1)). Hence this is not a rational cohomology complex quadric. So this case is uniqueup to essential isomorphism and such (G,M) will be constructed in Section 11.6.

11. Compact transformation groups on rational cohomology complex quadrics withcodimension one orbits.

All the pair (G,M) which has codimension one principal orbits are exhibited in thislast section.

11.1. (SO(2n + 1), Q2n).In this case M = Q2n and SO(2n + 1) acts on M through the canonical representation

to SO(2n + 2). Then there are two singular orbits S2n and Q2n-1. The principal orbit typeis RV2n+1;2

∼= SO(2n + 1)/SO(2n − 1).

Put Z2 =

{In+2,

(−1 0

0 In+1

)}. This group canonically acts on Qn and commutes

with the action of SO(n+1). (SO(n+1), Qn/Z2) has two singular orbits P2n(R) and Qn-1

and the principal orbit is RVn+1;2/Z2. From [Uch77] Section 9.6, such manifold (SO(n +

56

1),M) is unique up to essential isomorphism that is (SO(n+1),M) ' (SO(n+1), Pn(C)).Hence we get the following proposition

PROPOSITION 11.1. For n ≥ 3, Qn/Z2 ∼= Pn(C).

11.2. (SU(n + 1), Q2n).In this case M = Q2n and SU(n + 1) acts by the natural representation of SO(2n + 2)

that is

SU(n + 1) 3 A + Bi 7→(

A −B

B A

)∈ SO(2n + 2).

Then there are two singular orbits, both orbit types are Pn(C). The principal orbit type isSU(n + 1)/(SO(2)× SU(n − 1)).

For G = U(n + 1) we get a similar result.

11.3. (Sp(1)× Sp(m), Q4m-2), m ≥ 1.In this case M = Q4m-2 (n = 2m − 1) and the action of Sp(1) × Sp(m) on Hm is

defined by Axh where (h,A) ∈ Sp(1)×Sp(m) and x ∈ Hm. Then there is a representationρ : Sp(1)× Sp(m) → SO(4m) that is

ρ(h,A) =

h1Im h3Im −h2Im h4Im−h3Im h1Im −h4Im −h2Imh2Im h4Im h1Im −h3Im

−h4Im h2Im h3Im h1Im

X −Y Z −W

Y X −W −Z

−Z W X −Y

W Z Y X

where h = h1 + h2i + h3j + h4k ∈ Sp(1) and A = X + Yi + Zj + Wk ∈ Sp(m).Hence there is an action of Sp(1)×Sp(m) on Q4m-2 through the representation ρ. Then

there are two singular orbits S2 × Pm(C) and Sp(m)/(Sp(m − 2) × U(1)). The principalorbit type is Sp(1)×T1 Sp(m)/(Sp(m − 2)×U(1)).

11.4. (Spin(9), Q14).In this case M = Q14. It is well known that Spin(9) acts on S15 transitively by the

spin representation ρ : Spin(9) → SO(16) ([Yok73]). Hence Spin(9) acts on Q14 throughthis representation. Then the principal orbit type is Spin(9)/SU(3) ◦ T 1 and two singularorbits are Spin(9)/Spin(6) ◦ T 1 and Spin(9)/G2 ◦ T 1.

11.5. (G2, Q6).In this case M = Q6 and the exceptional Lie group G2 acts through the canonical

representation to SO(7). Then there are two singular orbits S6 and G2/S(U(1) × U(2)).The principal orbit type is RV7;2 ∼= G2/SU(2).

11.6. (S(U(3)×U(1)), Q4).In this case M = Q4. It is well known that there is the double covering representation

ρ : SU(4) → SO(6) ([Har90], [Yok73]) because of SU(4) ' Spin(6). Hence S(U(3)×U(1))acts on this manifold through the restricted representation of ρ, and the principal isotropy

57

group is S(U(1)×U(1))×S(U(1)×U(1)). Consequently principal orbits are of codimensionone.

11.7. (Sp(2), S7 ×Sp(1) P2(C)).In this case M = S7 ×Sp(1) P2(C) and Sp(2) canonical acts on S7 ∼= Sp(2)/Sp(1). The

manifold M is a quotient manifold of S7 × P2(C) by the action Sp(1) where Sp(1) acts onS7 ∼= Sp(2)/Sp(1) canonically and on P2(C) by the double covering Sp(1) → SO(3). Thenthe Sp(1) action on P2(C) has codimension one principal orbits Sp(1)/{1, −1, i, −i} andtwo singular orbits Sp(1)/U(1) and Sp(1)/U(1)j ∪U(1)ji where U(1)j = {a + bj| a2 + b2 =1}. Hence the Sp(2) action on M has codimension one principal orbits Sp(2)/Sp(1) ×{1, −1, i, −i} and two singular orbits Sp(2)/Sp(1)×U(1) and Sp(2)/Sp(1)×(U(1)j∪U(1)ji).

11.8. (G2 × T 1, GR(2,O)).In this case M = GR(2,O). Then g ∈ G2 acts u ∧ v ∈ M by g · u ∧ v = g(u) ∧ g(v). We

see g(u), g(v) is an oriented orthonrmal basis because of G2 ⊂ SO(7). Hence this actionis well defined on M. Moreover T 1 acts on M by the induced action from the canonicalSO(2)-action on O2. These two actions are commutative. Therefore we have the G2 × T 1-action on M.

Put G = G2× T 1. Then the isotropy subgroup G1^i is SU(3)× T 1, Gi^j is U(2)× T 1 andG1^1=

√2(i+j) is SU(2)◦T 1. Hence this action has codimension one orbit (G2×T 1)/SU(2)◦T 1

and two singular orbits (G2 × T 1)/(SU(3)× T 1) ∼= S6 and (G2 × T 1)/(U(2)× T 1).

58

Part 2

Equivarinat Graph Cohomology of Hypertorusgraph and (n + 1)-dimensional Torus action on

4n-dimensional manifold

12. Introduction of Part 2

A research in Part 2 is motivated by two problems about GKM-graphs and hypertoricmanifolds. First we mention a GKM-graph.

Let M2m be a 2m-dimensional manifold which has an n-dimensional torus action. Wedenote it by (M2m, Tn). This pair (M2m, Tn) is called a GKM-manifold if it satisfies thefollowing three conditions (GKM-condition);

• Its fixed point set MT is finite.• (M2m, Tn) is an equivariantly formal space.• (M2m, Tn) satisfies a pairwise linearly independence around its fixed point.

Here an equivariantly formal space (M2m, Tn) means the spectral sequence of the fibrebundle

M → ET ×T M → BT

collapses (see [GKM98]), and a pairwise linearly independence means the induced Tn-action on the tangent space of a fixed point Tp(M) is equivariantly decompose into V(α1)×· · · × V(αm) such that the weights {α1, . . . , αm} are pairwise linearly independent in t∗,where t∗ is a dual Lie algebra of the torus T .

A regular m-valent graph Γ(M)(= Γ) = (V` , E` ) can be defined by the above GKM-manifold (M2m, Tn), regarding the fixed point in MT as a vertex in V` and the connectedcomponent in the orbit space of one-dimensional orbits as an oriented edges E` . Moreover“labels” on the oriented edges E` are defined by its isotropy weight representations (inthe dual Lie algebra (tn)∗Z). We denote it α : E` → (tn)∗Z and call α an axial functionon Γ . The important fact in [CS74] and [GKM98] is that the equivariant cohomologyring of (M2m, Tn) is isomorphic to an equivariant graph cohomology of Γ(M) defined by(M2m, Tn) (see Section 13), that is the following equation holds;

H∗T (M;Z) ' H∗

T (Γ(M), α),

where H∗T (M;Z) is the equivariant cohomology of (M,T) and H∗

T (Γ(M), α) is the equivari-ant graph cohomology of the GKM-graph Γ(M).

Now a GKM-manifold (M2m, Tn) is a geometrical object, on the other hand, a GKM-graph Γ can be assumed to be a combinatorial object. So we are naturally led to study tocompute the equivariant cohomology ring H∗

T (M;Z)(' H∗T (Γ, α)) from combinatorial stru-

cures of Γ . In fact, it has already succeeded in some cases. In 2001, Guillemin and Zarainitiated to study a class of the GKM-graph Γ (a toric graph) which contains the GKM-graph defined by the toric manifold. They give generators of H∗

T (Γ) as H∗(BT)-module bycombinatorial structures of Γ and they compute the Betti number of H∗

T (Γ) in [GZ01]. In2003, Masuda and Panov study on a torus manifold which is more general than the toricmanifold in [MP03]. They compute its equivariant cohomology ring and describe it bycombinatorial structures of their characteristic manifolds. Maeda and they also define aclass of GKM-graphs (a torus graph) in [MMP05] which contains the GKM-graph defined

by the torus manifold and describe its graph cohomology ring by combinatorial struc-tures (by a connection θ of Γ which will be defined in Section 13). The torus manifold M

(contains the toric manifold) is a 2n-dimensional manifold which has a Tn-action and itsatisfies a GKM condition if M holds Hodd(M) = 0. So, in the above cases, the GKM-graphΓ is an n-valent graph and has an axial function α : E` → (tn)∗Z. However little is knownabout the GKM-graph Γ which is an m-valent and has an axial function α : E` → (tn)∗Zsuch that m > n, we call such GKM-graph an (m,n)-type GKM-graph. Therefore the firstmotivation of Part 2 is as follows.

PROBLEM 1. Let (Γ, α, θ) be an (m,n)-type GKM-graph, where m > n . Describe the graphcohomology ring H∗

T (Γ, α) by its combinatorial structures (a connection θ).

Next we mention a hypertoric manifold.In 2000 [BD00], Bielowski and Dancer define 4n-dimensional variety, the hypertoric

variety M4n, by the hyperKaler quotient of a torus action on the quaternionic spaces.Remark the hypertoric variety and the toric hyperKaler are same things, we use hypertoricvariety in Part 2. The hypertoric variety corresponds to a hyperplane arrangement. Inthe same year, Konno computed the equivariant cohomology ring (and the ordinary co-homology ring) of (M4n, Tn) by the combinatorial structure of hyperplane arrangements[Kon00] and [Kon03]. The hypertoric variety (M4n, Tn) satisfies the upper two condi-tions in GKM-condition, that is MT is finite and it is the equivariantly formal (becauseof Hodd(M) = 0). However it does not satisfy the pairwise linearly independentness onMT , that is, it does not satisfy the GKM-condition. Hence we can not define the GKM-graph from the hypertoric variety (M4n, Tn). In 2004 [HH04], Harada and Holm found ahypertoric variety (M4n, Tn) extends to a transformation group (M4n, Tn+1) and it satis-fies the GKM-condition. Therefore we can define the (2n, n + 1)-type GKM-graph from(M4n, Tn× S1). Moreover they describe the eqivariant cohomology ring of (M4n, Tn× S1)in terms of its hyperplane arrangement and they coorespond its generator to an elementof its equivariant graph cohomology H∗

T (Γ). So we are naturally led to think there mightbe a class of GKM-graphs which contains the GKM-graph defined by the hypertoric vari-ety like the Masuda-Maeda-Panov’s torus graph in [MMP05]. From the above researches,we can denote the second motivation as follows;

PROBLEM 2. Define the class of GKM-graphs which contains the GKM-graph coming fromthe hypertoric variety and compute its equivariant graph cohomology.

The aim of Part 2 is to solve the above Problem 2. We define a hypertorus graph asa new class of GKM-graphs, which contains the GKM-graph defined by the hypertoricvariety or the cotangentbundle of the torus manifold and describe its equivariant graphcohomology in terms of its combinatorial structure for some cases. The hypertorus graphis (2n, n + 1)-type GKM-graph, so to describe its equivariant graph cohomology is tosolve the Problem 1 partially. A main result of Part 2 is a generalization of the main

61

result in [HH04] from the other aspect, which is not the hyperplane arrangement but thehypertorus graph. We also define a quaternionic torus graph as generalization of hypertorusgraph. This graph contains the GKM-graph coming from a complex quadric or a quaternionprojective space. We do not compute its equivariant graph cohomology in Part 2, but tocompute it might be an important problem.

The main result of Part 2 is as follows.

MAIN THEOREM 2. Assume for each codimension two hypertorus subgraph L there is aunique hyperfacet H and its opposite side H such that ∂H = L, and H ∩ G = ∅ or connected forall hyperfacets H and G. Then there is the following isomorphism:

H∗T (Γ, α) ' Z[Γ, θ].

The organization of Part 2 is as follows. First we recall a GKM-graph and its equi-variant graph cohomology in Section 13. Because we would like to define a quaternionictorus graph which contains the GKM-graph coming from Tn+1-action on HPn, we definethe GKM-graph more general than the GKM-graph which is defined in other papers, forexample [GZ01] or [MMP05]. In Section 14 we define a hypertorus graph and quater-nionic torus graph, before to define we recall a hypertoric variety. Next we exhibit threeexamples of hypertorus graph and quaternionic torus graph in Section 15. To state ourmain theorem, we have to prepare some notations and propositions in Section 16. Finallywe prove the main theorem in Section 17. To prove the main theorem, we consider threecases as follows:

(1) Γ has only one vertex;

(2) Γ is a minimal hypertorus graph;

(3) Γ is general hypertorus graph.

13. GKM-graph and equivariant graph cohomology

Guillemin and Zara [GZ01] introduce a GKM-graph to study equivariant cohomologyrings of GKM-manifolds from combinatorial aspects. They succeed to translate the no-tations from toric manifolds into toric graphs, where a toric graph contains a GKM-graphwhich is defined by toric manifolds. For instance they define a Betti number of toricgraphs by its combinatorial information and show it accords with a Betti number of toricmanifolds.

Maeda, Masuda and Panov introduced the torus graph in [MMP05] as a GKM-graphwhich is more general than a toric graph. They succeed to describe its equivariant graphcohomology rings.

In this section we state a definition of a GKM-graph which has more general condition(of an axial function) than the above two papers [GZ01] and [MMP05] and also define itsequivariant graph cohomology.

62

Remark. A toric graphs called GKM-graphs in [GZ01], but we use the terminologyGKM-graph is more general meaning in Part 2.

First we state some notations. Let Γ = (V` , E` ) be a connected regular m-valent graphwhich is possible to have a leg, where V` is a set of finite vertices of Γ . Here E` is a setwhich consists of two parts as follows:

E` = E` ∪ L` .

Here E` is a set of all oriented edges, so each edge e ∈ E` has two possible orientations,and L` is a set of legs where a leg l ∈ L` is an out going half line from the vertex, so eachleg l has an only one orientation. The following two figures are examples of our graphs.

leg

edge edge

FIGURE 13.1. Examples.

The above left example is a 2-valent graph which has two legs and edges and the rightone is 3-valent graph which has no legs.

An opposite orientation of the edge e = pq is denoted by e = qp, we also denote theinitial vertex of e = pq by i(e)(= p) and the terminal vertex of e by t(e)(= q). So a leg l

does not have terminal vertex but it has an initial vertex i(l), hence we can state the leg l

is an out going half line from i(l). The leg has only one orientation.

Next we prepare two important notations, a connection and an axial function. We canregard a connection as a combinatorial structure on the graph Γ , on the other hand anaxial function as an algebraic structure on it.

For p ∈ V` we put

E`p = {e ∈ E` | i(e) = p},

and for an edge e = pq ∈ E` we denote a collection of bijections

θe : E`p → E`qby θ = {θe}. Now we denote the number of all edges and legs which have a same initialvertex p by |E`p |. In our case |E`p | = m holds for all p ∈ V` because the graph Γ is anm-valent graph. Hence the bijective map θe always exits on all edges E` . Let us state adefinition of a connection.

63

Definition[connection]. A connection on Γ is a collection θ = {θe} which satisfies thefollowing two conditions:

(1) θe = θ-1e ;

(2) θe(e) = e.

We can easily show an m-valent graph Γ admits different ((m − 1)!)g connections,where g is the number of (non- oriented) edges E` .

Next we define an axial function which is more general than the definition of axialfunctions in [GZ01] and [MMP05].

Definition[axial function]. We call a map α : E` → Hom(T, S1) = H2(BT) = tZ an axialfunction (associated with the connection θ) if it satisfies the following three conditions:

(1) mi(e)α(e) = mi(e)α(e) for some mi(e), mi(e) ∈ Z− {0};

(2) Elements of α(E`p) are pairwise linearly independence for each p ∈ V` ;

(3) m ′e ′α(θe(e

′)) − me ′α(e ′) ≡ 0 (mod α(e)) for any e ∈ E` , e ′ ∈ E`i(e) and somenon-zero integer m ′

e ′ , me ′ which depend on e ′.

We call the above third relation a congruence relation.Remark. The GKM-graph which defines in [GZ01] (resp. in [MMP05]) is the first

condition of the axial function was mi(e) = −1 = −mi(e) (resp. mi(e) = ±1 and mi(e) = 1)and the congruence relation was both of them were m ′

e ′ = me. Because we would liketo define a quaternionic torus manifold as a class of GKM-graphs which contains the GKM-graph defined by Tn+1-action on HPn, we need to define the axial function which hasmore general condition than [GZ01] and [MMP05].

Let us define a GKM-graph.Definition[GKM-graph]. Assume a connection θ defines on an m-valent graph Γ and

Γ is labeled by an axial function α whose terget is tnZ . Then we call (Γ, α, θ) a (m,n)-typeGKM-graph.

The GKM-graph is defined by a GKM-manifold as follows. Put vertices V` by MT ,edges and legs E` by the set of connected components of s, where s = {x ∈ M | dimT(x) =1}/T . Remark s is a one dimensional open manifold from the GKM-condition. Set thegraph Γ(M) by s, where s = {x ∈ M | dimT(x) = 1}

c/T = s ∪ V` . Then we call s a

one skelton of T -action on M (Xc means a closure of X). Since GKM-manifold satisfiesthe pairwise linerly independence around its fixed points, there is an isotropy weightdecomposition on the tangent space of p ∈ MT = V` as

Tp(M) ' V(α1)⊕ · · · ⊕ V(αn),

64

where αi ∈ (tm)∗ is a weight of an isotorpy group representation on Tp(M) and the rep-resentation space of αi is denoted by V(αi) ' C for all i = 1, · · ·n. Now each αi corre-sponding to some ei ∈ E` which has an initial point p.

Remark. We can assume ei as the Tm-1-invariant manifold in M, that is (Tei)c ' CP(1)

or Cwhich contains p ∈ MT .So an axial function αM on Γ(M) is the map which satisfies αM(ei) = αi. Finally we

define the connection θ from the above axial function αM (possibly not unique). Thereforewe get an (m,n)-type GKM-graph (Γ(M), αM, θ) from a GKM-manifold (M2n, Tm).

Guillemin and Zara define a toric graph in [GZ01]. The toric graph is an (n, n)-typeGKM-graph (without legs) and it satisfies mi(e) = −mi(e) = 1 on the first condition andm ′e ′ = me ′ = 1 on the third condition of the axial function. We call such axial function a

toric axial function. Moreover α(E`p) forms a basis of tnZ .Maeda, Masuda and Panov define a torus graph in [MMP05]. The torus graph is an

(n,n)-type GKM-graph (without legs) and it satisfies mi(e), mi(e) = ±1 on the first con-dition and m ′

e ′ = me ′ = 1 on the third condition of the axial function. We call such axialfunction a torus axial function. In this case α(E`p) also forms basis of tnZ . The torus graphcontains the toric graph and Maeda, Masuda and Panov show its equivariant graph co-homology is isomorphic to some ring which is defined by its combinatorial information.The following Figure 13.2 is an example of torus graph and the next one Figure 13.3 is anexample of generalized torus graph which is possible to have legs.

α

β

α−β

−α+β

−α

−β

p q

r

FIGURE 13.2. The GKM-graph associated with T 2-action on CP(2).

α

β −α+β

−αp q

FIGURE 13.3. The GKM-graph associated with T 2-action on CP(2) − {r}.

65

Give a GKM-graph (Γ, α, θ). Then we can define a ring H∗Tn(Γ, α) which is called an

equivariant graph cohomlogy.Definition[equivariant graph cohomology]. Let (Γ, α, θ) be an (m,n)-type GKM-

graph. Then we set an equivariant graph cohomology H∗Tn(Γ, α) as follows:

H∗Tn(Γ, α) = {f : V` → H∗

Tn(pt) | f(p) − f(q) ≡ 0 (mod α(pq))}

where pq ∈ E` is an edge.

As is well known H∗Tn(pt) is a polynomial ring Z[x1, · · · , xn] where xi ∈ H2

Tn(pt) = tnZ .Now we exhibit an example of an element of H∗

T (Γ, α).

Examples. Let Γ be a GKM-graph as the above Figure 13.2. Put f : V` → H∗T (pt) as

follows:

f(p) = α(2α + β), f(q) = 2αβ, f(r) = 2α2 + β(α − β),

where t2Z ' 〈α, β〉. Then we have f(p) − f(q) = 2α2 − αβ ≡ 0 (mod α), f(q) − f(r) =2α(β − α) − β(α − β) ≡ 0 (mod β − α) and f(r) − f(p) = −β2 ≡ 0 (mod − β). Hence themap f is an element of H∗

T (Γ).

14. Hypertoric variety and hypertorus graph

A hyperkahler quotient is defined by Hitchin, Karlhede, Lindstrom and Rocek in[HKLR87] as a quotient which constructs an hyperkahler manifold. In 2000, Bielowskyand Dancer study a special case of a hyperkahler quotient that is a hypertoric variety in[BD00]. A hypertoric variety is constructed by a hyperkahler quotient of torus action onT ∗CN ' HN (see Section 14.1) like a toric variety, which is defined by a kahler quotient oftorus action on CN. In same year [Kon00], H. Konno studies its cohomology ring struc-ture in detail. In 2004, Harada and Holm relates the hypertoric variety with the GKMtheory in [HH04]. In this section, we recall a hypertoric variety and define a hypertorusgraph.

14.1. hypertoric variety.A hypertoric variety is motivated to define a hypertorus graph in this thesis, note that

in the paper [Kon00] a hypertoric variety called a toric hyperkahler but we use the namehypertoric variety as [HH04]. Let us recall a hypertoric variety. Consider the natural torusgroup K(⊂ TN)-action on T ∗CN. Then we can define a hyperkahler moment map as follows:

µHK : T ∗CN —−→ t∗ ⊕ t∗Ci∗−→ k∗ ⊕ k∗C.

Here t∗ and k∗ are dual Lie algebras of TN and K, i∗ is an induced homomorphism fromthe inclusion i : k → t, and a map µ is defined by

µ(z,w) =1

2

N∑

i=1

(|zi|2 − |wi|

2)∂i ⊕N∑

j=1

zjwj∂j,

66

where z = (z1, · · · , zN) is a point of a base space CN, w = (w1, · · · , wN) is a point of a fibrespace and ∂1, · · · , ∂N are canonical basis of t∗. Take a regular value of (ν, 0) ∈ k∗⊕ k∗C, thena manifold µ-1

HK(ν, 0)(⊂ T ∗CN) has an almost free K(⊂ TN)-action. Hence the quotientspace µ-1

HK(ν, 0)/K = M4n is an orbifold and it has a (TN/K =)Tn-action. This orbifold iscalled a hypertoric variety. Moreover M4n has an induced residual S1-action from a scalermultiplication on fibres of T ∗CN. Therefore M4n has a Tn × S1-action and it satisfies aGKM-condition. The tangent space of its fixed point p has isotropy weight decompositionas follows:

Tp(M) = V(α1)⊕ · · · ⊕ V(αn)⊕ V(−α1 + x)⊕ · · · ⊕ V(−αn + x),

where t∗ ' 〈α1, · · · , αn〉 and s ' 〈x〉.

Remark. Tn-action on M4n does not satisfy a pairwise linerly independentness.

Hence there exists a GKM-graph coming from a hypertoric variety M4n with (Tn×S1)-action which has a properties, that the edges or legs consist n-pairs {e+

i , e-i } (i = 1, · · · , n)

in E`p and their axial function holds α(e+i ) + α(e-

i ) = x. From these properties, we willdefine a hypertorus graph as in next section.

The most essential example of the hypertoric variety is a cotangent bundle of com-plex projective space T ∗CP(n) which has a natural Tn-action and S1-action on fibres. Thefollowing Figure 14.1 is the GKM-graph coming from T 2 × S1-action on T ∗CP(2).

−α+β

α−β

β

−αα

α+x−α+β+x

−β+x α−β+x

−β+x−α+β+x

−β

FIGURE 14.1. The GKM graph associated with T 2 × S1-action on T ∗CP2.

14.2. Hypertorus graph.In the begining, we define a quaternionic torus graph as generalization of hypertorus

graph.Definition[quaternionic torus graph]. Let Γ = (V` , E` ) be a regular 2n-valent graph,

possibly with legs. Let (Γ, α, θ) be a (2n, n + 1)-type GKM-graph. Its axial function α

satisfies two conditions such that67

(1) α(e) = ±α(e)

(2) α(e ′) ≡ εe ′α(θe(e′)) (mod α(e)) for any e ∈ E` , e ′ ∈ E`i(e) and εe ′ = 1 or −1.

For each p ∈ V` , we can put E`p = {e+1 (p), · · · , e+

n(p), e-1 (p), · · · , e-

n(p)} and the pair(e+i (p), e-

i (p)) satisfies

α(e+i (p)) + α(e-

i (p)) = x(p)(14.1)

for all i = 1, · · · , n where an element x(p) ∈ (tn+1)∗ depends on p ∈ V` . Moreover the set

{α(e+1 (p)), · · · , α(e+

n(p)), x(p)}

is a basis of tn+1 for all p ∈ V` . Then we call such GKM-graph a quaternionic torus graph.

The following proposition can be proved by easy calculating.

PROPOSITION 14.1. Let (Γ, α, θ) be a quaternionic torus graph and x(p) ∈ t∗ be a value ofα(e+

i (p)) + α(e-i (p)) for each p ∈ V` , where {e+

i (p), e-i (p)} is a pair of E`p . Then the following

two statements are equivalent.(1) The edge pq ∈ E` satisfies θpq(e

+i ) = h+

i , θpq(e-i ) = h-

i for all i = 1, · · · , n andα(e) ≡ α(θpq(e)) (mod α(pq)) for all e ∈ E`p , that is εe = 1 for all e ∈ E`p .

(2) The equation x(p) − x(q) ≡ 0 mod α(pq) holds for the edge pq.

PROOF. First we show (1 ⇒ 2). Because Γ is a quaternionic torus graph, we have

α(e+i (p)) + α(e-

i (p)) = x(p) andα(e+

j (q)) + α(e-j (q)) = x(q),

for all i, j = 1, · · · , n. Now we can put θpq(e+(p)) = e+(q) and θpq(e

-(p)) = e-(q), andthen we have α(e+(p)) − α(e+(q)) ≡ 0 and α(e-(p)) − α(e-(q)) ≡ 0 (mod α(pq)) by theassumption 1. From the above equations, we have

(α(e+(p)) − α(e+(q))) + (α(e-(p) − α(e-(q))) = x(p) − x(q) ≡ 0 (mod α(pq)).

So we get (1 ⇒ 2).Next we show (1 ⇐ 2). Put the edge pq by e(= e+) and qp by h(= h+). First we

begin to show θe(e-) = h-. Now we have the following equations by the definition of

the quaternionic torus graph:

α(e) + α(e-) = x(p);

α(e-) − εeα(θe(e-)) ≡ 0 mod α(e).

Hence we have

α(e-) − εeα(θe(e-)) = x(p) − α(e) − εeα(θe(e

-))

≡ x(p) − εeα(θe(e-)) ≡ 0 (mod α(h)),

68

by α(e) = ±α(h). Therefore we get εeα(θe(e-))−x(q) ≡ 0 (mod α(h)) by the assumption

2. So we get

εeα(θe(e-)) − rα(h) = x(q)

= α(h) + α(h-)

for some r ∈ Z. Because of the definition of quaternionic torus graph, we have εe = 1,r = −1 and θe(e

-) = h-. Therefore for (E`p 3)e+i , e-

i 6= pq, we see θpq(e+i ) = h1, θpq(e

-i ) =

h2 ∈ E`q are not equal to the pair of qp.Next we show α(h1) + α(h2) = x(q). Now we have

α(e+i ) + α(e-

i ) = x(p),

x(p) − x(q) ≡ 0 mod α(pq),

α(e+i ) − ε1α(h1) ≡ 0 mod α(pq),

α(e-i ) − ε2α(h2) ≡ 0 mod α(pq)

for some ε1, ε2 = 1 or −1. Therefore we have the following equation:

x(p) − ε1α(h1) − ε2α(h2)

≡ x(q) − ε1α(h1) − ε2α(h2) ≡ 0 mod α(pq).

So we have ε1 = ε2 = 1 and α(h1) + α(h2) = x(q) because of the definition of thequaternionic torus graph. ¤

The quaternionic torus graph contains the GKM-graph coming from a Tn+1-action ona quarternionic projective spaceHP(n) (see next section). Let us define a hypertorus graph.

Definition[hypertorus graph]. Let Γ = (V` , E` ) be a regular 2n-valent graph, possiblywith legs. We say (2n, n + 1)-type GKM-graph (Γ, α, θ) is a hypertorus graph if α is a torusaxial function, that is mi(e), mi(e) = ±1 and m ′

e ′ = me ′ = 1 on the definition of the axialfunction, and it satisfies

α : E` → (tn × s1)∗Z = 〈α1, · · · , αn〉Z × 〈x〉Zwhere the αi (i = 1, · · · , n) is a basis of (tn)∗ and x is a basis of (s1)∗ and (tn × s1)∗Z is theweight lattice in the dual Lie algebra of Tn × S1. Moreover for all p ∈ V` we can put

E`p = {e+1 (p), · · · , e+

n(p), e-1 (p), · · · , e-

n(p)}

and its axial function satisfies

α(e+i (p)) + α(e-

i (p)) = x,(14.2)〈α(e+

1 (p)), · · · , α(e+n(p)), x〉Z = 〈α1, · · · , αn〉Z × 〈x〉Z(14.3)

for all i = 1, · · · , n and vertices.

We see easily 〈α(e-1 (p)), · · · , α(e-

n(p)), x〉Z = 〈α1, · · · , αn〉Z × 〈x〉Z .The following corollary is easy to show by Proposition 14.1.

69

COROLLARY 14.1. Let (Γ, α, θ) be a hypertorus graph and θpq(e+) = h1 and θpq(e

-) = h2,where e+, e- are pair of E`p and h1, h2 are elements in E`q . Then we have h1 = h+ and h2 = h-,that is h1 and h2 consist the pair in E`q .

We will exhibit examples in the next section.

15. Typical examples

In this chapter, we exhibit some examples which define a hypertorus graph and aquaternionic torus graph as a GKM-graph. First example is about a hypertorus graphand second and third examples are about a quaternionic torus graph.

15.1. cotangent bundle of torus manifold.The torus manifold is a 2n-dimensional compact smooth manifold M with an effective

action of an n-dimensional torus T whose fixed point set (finite) is non-empty. A character-istic submanifold of M is a codimension-two connected component of the fixed pointwiseby a circle subgroup of T . An omniorientation of M consists of a choice of orientation forM and for each characteristic submanifold. The torus manifold defined by Masuda in[Mas99] and [HM03]. It contans the toric manifolds and it satisfies a GKM-condition ifHodd(M) = 0. So we get a GKM-graph as a torus graph from torus manifold.

Denote a cotangent bundle of a torus manifold M by T ∗M. Then T ∗M has a canonicalTn-action and the scaler S1-action on fibres. Of course this case also satisfies a GKM-condition, so we have a GKM-graph. Moreover we can easily show this graph is a hyper-torus graph.

One of the example of torus manifolds (but not toric manifolds) is 2n-dimensionalsphere S2n (n ≥ 2). This manifold S2n(⊂ Cn × R) has a Tn-action ρ coming from thecanonical Tn-action on Cn. Define Tn × S1-action on T ∗S2n by the Tn-action induced fromthe above action ρ and the scaler S1-action on fibres. Let Γ = (V` , E` ) be the graph givenby finite fixed points and the one skelton of the orbit space. In this case there are just twofixed points that is V` = {N,S}, n edges connecting two vertices N, S and each vertex hasn legs, that is Γ has 2n legs. Moreover we can get the axial function α by the isotropyweight representation on fixed points and the connection θ is defined by this functionα. Then (Γ, α, θ) is a hypertorus graph. The following Figure 15.1 is a hypertorus graphassociated with T 2 × S1-action on T ∗S4.

15.2. quaternionic projective space.The quaternionic projective spaceHP(n) is a 4n-dimensional projective space over the

quaternionic numbers which defines as follows:

HP(n) = (Hn+1 − {0})/H∗,where H is the quaternionic numbers and H∗ is H− {0}. Remark the scaler multiplicationof H∗ on Hn+1 − {0} by the right side.

70

β α

−α+x −β+x

α

−β+x−α+x

β

FIGURE 15.1. The hypertorus graph associated with T 2 × S1-action on T ∗S4.

Then (n + 1)-dimensional torus Tn+1 acts HP(n) as follows:

(t1, · · · , tn, tn+1) · [h0 : h1 : . . . : hn] = [t1=2

n+1h0 : t1=2

n+1t1h1 : . . . : t1=2

n+1tnhn],

where (t1, · · · , tn, tn+1) ∈ Tn+1 and [h0 : h1 : . . . : hn] ∈ HP(n). Note that the left diagonalaction of tn+1 ∈ S1 on HP(n) is not trivial because the scaler H∗ acts from right side.

Then this action defines a GKM-graph Γ . This graph Γ is not a hypertorus graph but aquaternionic torus graph.

The following Figure 15.2 is a quaternionic torus graph coming from T 3-action onHP(2).

α+γ

α

β

β+γ −α

α+γ

α+β+γ

−α+β

α−β

α+β+γ

−β

β+γ

FIGURE 15.2. The quaternionic torus graph associated with T 3-action on HP2.

71

15.3. complex quadric.The complex quadric Q2n is a non-degenarate degree two homogeneous space in the

(2n + 1)-dimensional complex projective space P2n+1(C) which is defined as

Q2n = {z ∈ P2n+1(C) | z1z2 + · · ·+ z2n+1z2n+2 = 0}

where z = [z1 : . . . : z2n+2] ∈ P2n+1(C). This manifold Q2n has an (n + 1)-dimensionaltorus Tn+1 action as follows:

(t1, · · · , tn+1) ◦ [z1 : . . . : z2n+2] = [t1z1 : t-11 z2 : . . . : tn+1z2n+1 : t-1

n+1z2n+2],

where (t1, · · · , tn+1) ∈ Tn+1. Then this action satisfies the GKM-condition. It has 2n + 2

fixed points and the axial function α(pq) = −α(qp). The shape of graph is the completegraph except all diagonal edges. The following Figure 15.3 is the quaternionic torus graphcoming from T 3-action on Q4.

−α+β

−β+α −α+γ −α−β

−α−γ

−β−γ

β−α−β+γ

α+γ

β+γ

−α+γ −β+γ

β−γβ+α

β+γ

β−α

α−β

α−γα+β

α+γ

−α−γ

−β−γ

β−γ −γ+α

FIGURE 15.3. The quaternionic torus graph associated with T 3-action on Q4

In next section we will state a main theorem.72

16. Equivariant graph cohomology of hypertorus graph—Main theorem and Preparation—

In this section we state a main theorem of Part 2. To state a theorem, we prepare someterminologies. From now on (Γ, α, θ) means a hypertorus graph. First we give a definitoinof a pre-hyperfacet.

Definition[pre-hyperfacet]. Put a subgraph H = (VH, EH) ⊂ Γ = (V` , E` ) such that|EHp | = 2n − 1 or 2n for all p ∈ VH, where EHp = E`p ∩ EH and |E`p | means a number of outgoing edges and legs on p ∈ VH in H. Moreover this H is closed by a connection on Γ , thatis θpq|H : EHp → EHq is bijective (if |EHp | = |EHq |) or injective (if |EHp | < |EHq |) and if |EHp | < |EHq |

then θpq|H satisfies

α(e) − α(θpq(e)) = 0 (mod α(pq)) and

α(h) − x = 0 (mod α(pq))

where h ∈ E`q is an element which is not in Imθpq|H (h 6∈ Imθpq|H). We call H a pre-hyperfacet.

If the edge pq ∈ EH satisfies |EHp | < |EHq |, then the element of E`p − EHp denotes by nH(p)and we call it a normal edge or leg of H on p . The following proposition is easy to show bythe definition.

PROPOSITION 16.1. The normal edge (or leg) nH(p) stisfies θpq(nH(p)) 6∈ Im θpq|H forpq ∈ EHp which satisfies a number of edges (or legs) |EHq | = 2n.

PROPOSITION 16.2. Let nH(p) = e+ be a normal edge (or leg) of a pre-hyperfacet H. If thevertex q ∈ VH satisfies the assumption of Proposition 16.1, then we have pq = e-.

PROOF. From Proposition 16.1, we see θpq(e+) = h+ 6∈ Imθpq|H. Hence we have

α(h+) − x = kα(pq)

for some integer k by the definition of pre-hyperfacet. Since our GKM-graph is a pairwiselinearly independent on q and the equations α(h+)+α(h-) = x and α(pq) = ±α(qp), weget qp = h- and k = 1 or −1. By the equation θpq(e

+) = h+ and the congruence relationon pq, the following equation holds for some integer k ′:

α(e+) − α(h+) = k ′α(pq).

Hence we get α(e+) − (k + k ′)α(pq) = x from the above equations. Because our GKM-graph is a pairwise linearly independent on p and α(e+) + α(e-) = x, we have

α(pq) = α(e-) and k + k ′ = −1.

Hence we have pq = e-. ¤73

Next we start to mention generators of the equivariant graph cohomology H∗T (Γ, α).

Definition[Thom class of pre-hyperfacet]. Define τH : V` → H2(BT) by

τH(p) =

0 p 6∈ VH

x |EHp | = 2n

α(nH(p)) |EHp | = 2n − 1.

We call τH a Thom class of the pre-hyperfacet H.

Then we have τH ∈ H∗T (Γ, α) from the following proposition.

PROPOSITION 16.3. A Thom class τH is an element of an equivariant graph cohomologyH∗T (Γ, α).

PROOF. A Thom class is a map τH : V` → H2(BT), so we check this map satisfies theconditions of an element in H∗

T (Γ, α) (the congruence relation). In the case |EHp | = |EHq | = 2n

(pq is an edge), we have τH(p) − τH(q) = x − x = 0. So this map satisfies the congruencerelation on pq if |EHp | = |EHq | = 2n, that is τH(p) − τH(q) ≡ 0(mod α(pq)). Because thepre-hyperfacet H is closed by connection θ of Γ , we have the congruence relation even if|EHp | = |EHq | = 2n − 1.

If |EHp | < |EHq | (resp. |EHp | > |EHq |), then we have τH(p)−τH(q) = α(e ′)−x (resp. x−α(e ′)).From Proposition 16.2, the equation α(e ′) − x = −α(pq) holds. Hence a Thom class τHsatisfies the congruence relation for all edges. Therefore τH ∈ H∗

T (Γ). ¤Thom classes will be generators of H∗

T (Γ, α).

Next we define an opposite side of pre-hyperfacet to except a Thom class associatedwith a disconnected pre-hyperfacet from genarators of H∗

T (Γ, α).Definition[opposite side of pre-hyperfacet]. If a pre-hyperfacet H satisfies the fol-

lowing:

τH + τH = x

for a pre-hyperfacet H, then we call H an opposite side of H.

The following proposition holds for the opposite side of the pre-hyperfacet.

PROPOSITION 16.4. For all pre-hyperfacet H in the hypertorus graph (Γ, α, θ), there is aunique opposite side H and the opposite side H is a pre-hyperfacet.

PROOF. Take a pre-hyperfacet H = (VH, EH) in Γ which have p ∈ VH such that |EHp | =

2n−1. We construct H as follows. If the vertex q ∈ VH has 2n out going edges (or legs) inH that is EHp = E`p , then we put p 6∈ VH and E Hp = ∅. If the vertex q is not in VH (q 6∈ VH),then we put q ∈ VH and E Hq = E`q . If the vertex r ∈ VH has 2n − 1 edges or legs in H andEHr = {e+

1 , · · · , e+n , e-

1 , · · · , e-n-1}, then we set r ∈ VH and E Hr = {e+

1 , · · · , e+n-1, e

-1 , · · · , e-

n}.The above H = (VH, E H) is closed under the connection θ|E from Proposition 16.2. So this

74

pre-hyperfacet H is an opposite side of H, that is τH+τH = x, from the above constructionand uniqueness is easy to show. ¤

The following Figure 16.1 is one of the pre-hyperfacet and its opposite side in theexample of Figure 14.1, the value on each vertex is the value of its Thom class.

α−β

α 0

−α+β+x

−α+x

x

FIGURE 16.1. Thom class of pre-hyperfacet and its opposite side.

Here we state a generator of the equivariant graph cohomology of Γ .Definition[hyperfacet]. We call a connected pre-hyperfacet a hyperfacet if its opposite

side is connected.

The following Figure 16.2 is an example which is not a hyperfacet but a pre-hyperfacet.

pre-hyperfacet its opposite sideFIGURE 16.2

By the definition of hyperfacet and Proposition 16.4, we have the following proposi-tion.

PROPOSITION 16.5. For the hyperfacet H, its opposite side H is the hyperfacet.

We prepare the following notation.Definition[boundary of hyperfacet]. We denote ∂H = H ∩ H where H is a hyperfacet

and we call ∂H a boundary of hyperfacet.75

From the connection of hyperfacet, we have the following proposition.

PROPOSITION 16.6. ∂H = (V@H, EH∩E H) is a codimension two ((2n−2)-valent) hypertorussubgraph.

For the codimension two hypertorus subgraph, the following proposition holds.

PROPOSITION 16.7. For the vertex p ∈ V` and the edge (or leg) e ∈ E`p , there is a uniquecodimension two hypertorus subgraph Γ ′ = (V`

′, E` ′) whose normal edge (or leg) on p is e.

PROOF. Put e = e+ ∈ E`p . Let {e+, e-} be a pair in E`p . Then we can construct a (2n−2)-valent hypertorus subgraph Γ ′ such that E` ′p = E`p − {e+, e-} as follows.

First we take n−1 lines via p which contain E` − {e+, e-} = E` ′p . We denote an abstructgraph which is defined by these lines by Lp. Take q ∈ Lp such that pq ∈ E` . Then we cantake E`q − {θpq(e

+), θpq(e-)} = E` ′q . From Corollary 14.1, E` ′q consists of n − 1 pairs in E`q

and the restricted bijection θpq|E` ′p : E` ′p → E` ′q is well-defined. Next we take n−1 lines viaq which contain E` ′q (and denote it by Lq). Similarly we can get a graph Γ1 = ∪q∈VLpLq.

If this graph Γ1 is (2n − 2)-valent graph then we get a codimension two hypertorussubgraph that we want. Assume this graph Γ1 has a vertex r which is not (2n − 2)-valent.Then there is a path l from p to r. Denote the edge (or leg) in E`r which corresponds to e+

by θl(e+). If we can take two diffenrent paths l1, l2 from p to r. Then we have θl1(e

+) =θl2(e

+) or {θl1(e+), θl2(e

+)} is a pair in E`r , because of Corollary 14.1, the congruencerelation and the definition of the hypertorus graph that if we put {e+

1 , · · · , e+n} ⊂ E`p , then

〈x, α(e+1 ), · · · , α(e+

n)〉 ' tZ for all p. Hence we get a (2n−2)-valent hypertorus subgraphΓ ′ ⊂ Γ to apply the similar argument. ¤

Before to state a main theorem, we prepare a notation.Notation. Let (Γ, α, θ) be a hypertorus graph. Denote the set of all hyperfacet of Γ by

H. The algebra Z[Γ, θ] is as follows:

Z[Γ, θ] = Z[x, H | H ∈ H]/I,

where Z[x, H | H ∈ H] is a polynomial ring generated by all hyperfacets of Γ , x, and theideal I is generated by

H + H − x for all H ∈ H and∏

H∈H ′H where H ′ ⊂ H is the set

⋂

H∈H ′H = ∅.

Let us state a main theorem.

MAIN THEOREM 2. Assume for each codimension two hypertorus subgraph L there is aunique hyperfacet H and its opposite side H such that ∂H = L, and H ∩ G = ∅ or connected forall hyper facets H and G. Then there is the following isomorphism:

H∗T (Γ, α) ' Z[Γ, θ].

76

Before to show the above theorem we prepare notation.Definition[neighborhood of subgraph]. Let H be a subgraph of Γ . Put N(H) be a

2n-valent graph in Γ which satisfies the following properties:

VN(H) = VH;

EN(H)p = EHp if |EHp | = 2n;

EN(H)q = EHq ∪ {l(n(q)1), · · · , l(n(q)k)} if |EHq | = 2n − k,

where {n(q)1, · · · , n(q)k} = E`q − EHq . Here if n(q) is a leg then l(n(q)) = n(q), if not sothen we regard the edge n(q) as a leg whose initial vertex is q (denote it by l(n(q))). Wecall N(H) a neighborhood of the subgraph H in Γ .

Remark. We do not call a neighborhood N(H) a subgraph of Γ if N(H) has a leg l(n(q))such that n(q) is an edge in Γ . Of course the neighborhoods N(H) is a hypertorus graphfor every hyperfacet H.

The following figure is an image of the neighborhood of pre-hyperfacet in Figure 14.1.The upper image is an example whose neighborhood is not a subgraph in Γ .

pre-hyper facet its neighborhood

FIGURE 16.3. Hyperfacet and its neighborhood.

From the next section we will prove the main theorem.

17. Proof of the main theorem

In this section we show the main theorem. The program of proof is first we will provethe case |V` | = 1, and next we will prove about a minimal hypertorus graph by the in-ductive argument for |V` |, finally we will prove the general hypertorus graph by the in-ductive argument and the Mayer-Vietoris analogue dividing a non minimal hypertorusgraph into two hypertorus graphs.

77

17.1. The case |V` | = 1.First of all we prove Theorem 17.1 about the easiest hypertorus graph, that is

Γ = ({p}, {e+1 , · · · , e+

n , e-1 , · · · , e-

n}),

where e+i and e-

i are legs for all i = 1, · · · , n.Remark. If the quaternionic torus graph Γ has only one vertex (|V` | = 1), then Γ is always

the above graph. So the following theorem also holds on the quaternionic torus graph.

THEOREM 17.1. Set the hypertorus graph Γ = ({p}, E` ), that is the graph consists of only onevertex and 2n legs. Then we have Z[Γ, θ] ' H∗

T (Γ, α).

PROOF. We can put E` = {e+1 , · · · , e+

n , e-1 , · · · , e-

n}. By the definition of hypertorusgraph, we have 〈α(e+

1 ), · · · , α(e+n)〉 ' tnZ and α(e-

i ) = x − α(e+i ) for all i = 1, · · · , n. Put

αi = α(e+i ). Then we have

H∗T (Γ, α) = {f : {p} → H∗

T (pt)} ' H∗T (pt) = H∗(BT) = Z[x, α1, · · · , αn]

where 〈x, α1, · · · , αn〉 = tZ.Let H be all hyperfacets in Γ . Then we can put H = {H1, · · · , Hn, H1, · · · , Hn}

such that τHi(p) = αi (that is nHi(p) = e+i and nHi(p) = e-

i ) from the definition of thehyperfacet and |V` | = 1. Since the intersection of all hyperfacets is ∩H∈H = {p}, we have

Z[Γ, θ] = Z[x, H | H ∈ H]/〈H + H − x | H ∈ H〉.By the above ideal 〈H+ H−x | H ∈ H〉 = 〈Hi+ Hi−x | i = 1, · · · , n〉 = I, we can assume[Hi] = [x − Hi] in Z[Γ, θ], so we get the natural surjective homomorphism

ϕ : Z[x, H1, · · · , Hn] → Z[Γ, θ].

Next we put

ρ : Z[Γ, θ] → Z[x, H1, · · · , Hn]

by ρ([x]) = x, ρ([Hi]) = Hi and ρ([Hi]) = x − Hi. If [X] = [Y] ∈ Z[Γ, θ] then X − Y ∈ I ⊂Z[x, H | H ∈ H], that is X − Y = Z(Hi + Hi − x) for some i and Z ∈ Z[x, H | H ∈ H].Hence we have ρ([X]) = ρ([Y]) from ρ([X − Y]) = ρ([Z])(Hi+ ρ([Hi]) − x) = 0. So this mapρ is a well-defined homomorphism. Because the composite map is ρ ◦ ϕ = id from thedefinition of ρ, we have ϕ is an isomorphism and ρ = ϕ-1. So we get

Z[Γ, θ] ' Z[x, H1, · · · , Hn].

Hence the following map is isomorphic:

Ψ : Z[Γ, θ]−→ Z[x, H1, · · · , Hn]

−→ Z[x, α1, · · · , αn] ' H∗T (Γ, α)

such that ψ(x) = x and ψ(Hi) = τHi(p) = αi. Therefore we have Ψ([x]) = x, Ψ([Hi]) = τHiand Ψ([Hi]) = x − τHi = τHi . ¤

Next we show our main theorem about the subclass (minimal hypertorus graphs) of thehypretorus graphs.

78

17.2. The case where Γ is the minimal hypertorus graph.A minimal hypertorus graph is a hypertorus graph which can not divide into two

hypertorus graphs. The rigorous definition is as follows.Definition[minimal hypertorus graph]. Let Γ be a hypertorus graph. We put a set of

all hyperfacets in Γ as follows:

H = {H1, · · · , Hm, H1, · · · , Hm}.

Then we call Γ is a minimal if it satisfies the neighborhood of Hi coincides with Γ , that isN(Hi) = Γ for all i = 1, · · · ,m.

In this section, we show the main theorem on all minimal hypertorus graphs.

17.2.1. Injectivity.First we show the following lemma.

LEMMA 17.1. Let Γ be a minimal hypertorus graph and H = {H1, · · · , Hm, H1, · · · , Hm}

is a set of all hyperfacets in Γ such that N(Hi) = Γ for all i = 1, · · · ,m. Then there is the followingisomorphism:

Z[Γ, θ] = Z[x, H1, · · · , Hm, H1, · · · , Hm]/I' Z[x, H1, · · · , Hm]/〈

∏

H∈H ′H |H ′ ⊂ {H1, · · · , Hm}〉,

where H ′ is a set of disjoint hyperfacets in {H1, · · · , Hm}.

PROOF. Because the relation N(Hi) = Γ holds, we have V` = VHi for all i = 1, · · · , m.So we see that if Hi ∩ (∩lj=1Hj) = ∅, then ∩lj=1Hj = ∅. Hence we can put

I = 〈∏

H∈H ′H, H + H − x〉

such that H ′ ⊂ {H1, · · · , Hm}.Put the ideal I ′ as follws:

I ′ = 〈∏

H∈H ′H〉.

Because we have [Hi] = [x − Hi] for all i = 1, · · · , m in Z[Γ, θ], the following map iswell-defined and surjective:

ϕ : Z[x, H1, · · · , Hm]/I ′ → Z[x, H1, · · · , Hm, H1, · · · , Hm]/Isuch that ϕ((x)) = [x], ϕ((Hi)) = [Hi].

Moreover the following map ρ is an inverse map of ϕ:

ρ : Z[x, H1, · · · , Hm, H1, · · · , Hm]/I → Z[x, H1, · · · , Hm]/I ′such that ρ([x]) = (x), ρ([Hi]) = (Hi), ρ([Hi]) = (x − Hi). Hence ϕ is an isomorphism.Therefore we get Z[Γ, θ] ' Z[x, H1, · · · , Hn]/I ′. ¤

79

Next we put the homomorphism Ψ : Z[Γ, θ] → H∗T (Γ, α) as follows:

Ψ([x]) = x;

Ψ([H]) = τH.

Then this map is well-defined by the following equations:

τH + τH = x;∏

H∈H ′τH = 0,

where H ′ is a set in H such that the intersection of all elements is ∩{H ∈ H ′} = ∅. Thefollowing lemma holds for this map Ψ.

LEMMA 17.2 (Injectivity). Let Γ be a minimal hypertorus graph. If there exists unique hyper-facet H and its opposite side H such that ∂H = L for every codimension two hypertorus subgraphL, then Ψ is injective.

PROOF. Define ψ : Z[x, H1, · · · , Hm]/I ′ → H∗T (Γ, α) as follows:

ψ([x]) = x

ψ(Hi) = τHi.

Then we have ψ ◦ ρ = Ψ. So the injectivity of Ψ is equivalent to the injectivity of ψ. Wewill prove the injectivity of ψ.

Put Z[Γ ]p = Z[x, H1, · · · , Hm]/〈H | p 6∈ VH〉. Then we have

Z[Γ ]p ' Z[x, H | p ∈ VH].

Put the homomorphism ψp : Z[x, H | p ∈ VH] → H∗T (pt) as follows:

ψp(x) = x(p),

ψp(H) = τH(p).

Now the hypertorus graph Γ is minimal and the opposite side of the generator H ofZ[x, H | p ∈ VH] is N(H) = Γ . So we have if p ∈ VH then p ∈ V@H. Since the axialfunctions around of the vertex p and x span t and we have Proposition 16.7 and the as-sumption of this lemma, we have this map ψp is isomorphic that is there is the followingisomorphism:

Z[Γ ]p ' Z[x, H | p ∈ V@H] ' H∗T (pt).

We can put χp : Z[x,H1, · · · , Hm]/I ′ → Z[Γ ]p by the canonical surjection because of I ′ ⊂〈H | p 6∈ VH〉. So we have the following commutative diagram:

Z[x, H1, · · · , Hm]/I ′ ffl−→ ⊕p∈V`Z[Γ ]pψ ↓ ↓'

H∗T (Γ, α)

ffi−→ ⊕p∈V`H∗T (pt),

80

where χ = ⊕p∈V`χp and φ(f) = ⊕p∈V` f(p). Because φ is injective, if the injectivity of χ isknown then we have the injectivity of ψ.

Since we have Ker(χp) = 〈H | p 6∈ VH〉/I ′, the followings hold:

Ker(χ) = Ker(⊕p∈V`χp)= ∩p∈V`Kerχp= ∩p∈V` (〈H | p 6∈ VH〉/I ′)= (∩p∈V` 〈H | p 6∈ VH〉)/I ′.

Hence we assume χ([X]) = 0 for some element [X] ∈ Z[x, H1, · · · , Hm]/I ′, then we have

[X] ∈ (∩p∈V` 〈H | p 6∈ VH〉)/I ′.Assume X ∈ ∩p∈V` 〈H | p 6∈ VH〉 ⊂ Z[x, H1, · · · , Hm]. Then we can denote uniquely by

X =∑

a; a1; ··· ; amk(a; a1; ··· ; am)x

aHa11 · · ·Ham

m

for some k(a; a1; ··· ; am) ∈ Z, because there is no relation on xaHa11 · · ·Ham

m and xa′Ha ′11 · · ·Ha ′m

m

in Z[x, H1, · · · , Hm] if (a, a1, · · · , am) 6= (a ′, a ′1, · · · , a ′m). Because of X ∈ 〈H | p 6∈VH〉, we have there is a hyperfacet Hj such that p 6∈ VHj and aj 6= 0 for each termk(a; a1; ··· ; am)x

aHa11 · · ·Ham

m of X. This fact holds for all p ∈ V` because X ∈ ∩p∈V` 〈H | p 6∈VH〉. Hence there are j1, · · · , jr such that∩rs=1Hjs = ∅ and aj1 , · · · , ajs 6= 0 for each term ofX. This means X ∈ I ′. Hence we assume χ([X]) = 0 then [X] = 0 in Z[x, H1, · · · , Hm]/I ′.So we have the injectivity of χ. ¤

17.2.2. Surjectivity.Next we prepare the following concept of the hypertorus graph.Definition[line and end point]. Let Γ be a hypertorus graph. We call the 2-valent

hypertorus subgraph l in Γ a line. We call an end point of the line l such that E lp has a leg.We also call an end point of Γ such that p is an end point of all line which through on p.

For the end point of the minimal hypertorus graph Γ , we have the following proper-ties.

LEMMA 17.3. Let Γ be a minimal hypertorus graph which satisfies the condition there is aunique pair hyperfacet H and its opposite side H for all codimension two hypertorus subgraph inΓ , then all the elementes of V` are end points of Γ .

PROOF. From Proposition 16.7 and an assumption of this Lemma, we can take Hi

(i = 1, · · · , n) such that N(Hi) = Γ for a vertex p ∈ V` as it has a normal edge e ∈ E`p .Now the normal edge e on p of Hi is a leg because of VHi = V` . Hence p is an end pointof all lines through on p. So the vertex p is an end point of Γ . ¤

Moreover we have the following proposition for the hypertorus graph which is inLemma 17.3

81

PROPOSITION 17.1. Let Γ be a hypertorus graph. If all vertices are end points of Γ ;, then thenumber of vertices on l is |V l| ≤ 2 for all lines l ⊂ Γ .

PROOF. Assume there is a line l such that |V l| ≥ 3. Then there is a vertex p ∈ V l whichis not an end point for this line l. Hence this is a contradiction. ¤

The following lemma will be used to prove the surjectivity of Ψ : Z[Γ, θ] → H∗T (Γ, α) for

the minimal hypertorus graph Γ . First we prove the following proposition.

PROPOSITION 17.2. Let Γ be a graph. Then there exists a vertex p ∈ V` such that Γ − N(p)is connected.

PROOF. We show the statement by the inductive argument. If the number of vertices|V` | = 2, then we easily see this proposition. Assume the statement of this propositionholds for all Γ such that |V` | < k. When the number of vertices |V` | = k, take a vertexp ∈ V` . If Γ −N(p) is not connected, then we can put Γ −N(p) = Γ1 ∪ Γ2. Because we have|V`1 | < |V` | = k and the assumption of the induction, there exists q ∈ V`1 ⊂ V` such thatΓ1 − N(q) is connected. Hence Γ − N(q) is connected. ¤

LEMMA 17.4. Let Γ be a hypertorus graph. There is a vertex p ∈ V` such that L − N(p) ∩ L

is connected for all codimension two hypertorus subgraph L.

PROOF. If |V` | = 2, then we can easily show this lemma. Assume this statement holdsall hypertorus graph Γ such that |V` | ≤ k − 1. If we put |V` | = k, we can take p ∈ V`

such that Γ ′ = Γ −N(p) is connected from Proposition 17.2. Then Γ ′ is a hypertorus graphwhich has k − 1vertices. So from our assumption, there is a vertex q ∈ V`

′ ⊂ V` such thatΓ ′ − N(q) satisfies the statement of this lemma.

Now we denote the codimension two hypertorus subgraph in Γ ′ by L ′. Then there isa codimension two hypertorus subgraph L of Γ such that L ′ ⊂ L. If p 6∈ VL, then L = L ′. Ifnot so then there are two cases where

(1) L ′ = L − L ∩N(p) is connected

(2) L − L ∩N(p) is a disjoint union L ′ ∪ L ′′.In each above case, (L ′−N(q)∩L ′)∪(L∩N(p)) is connected. Because N(q)∩L ′ = N(q)∩L

and p 6= q, we have (L ′−N(q)∩L ′)∪(L∩N(p)) = L−N(q)∩L. Hence we have L−N(q)∩L

is connected for all codimension two hypertorus graph L in Γ . ¤

Let us prove the surjectivity.

LEMMA 17.5 (Surjectivity). Let Γ be a minimal hypertorus graph. If it holds H1 ∩H2 = ∅ orconnected for all hyperfacets H1 and H2 in Γ and there is a unique pair {H, H} such taht ∂H = L

for every codimension two hypertorus subgraph L, then Ψ is surjective.

82

PROOF. We only show the surjectivity of ψ because of Ψ = ψ◦ρ. There is the followingcommutative diagram:

Z[x,H1, · · · , Hm]p−→ Z[x,H1, · · · , Hm]/I ′

π ↓ ↓ ψ

H∗T (Γ, α) = H∗

T (Γ, α)

where the natural projection p is surjective. So we will show that π is surjective by theinductive argument for |V` |.

If |V` | = 1, then we have this lemma by Theorem 17.1, hence π is surjective. Assumethe surjectivity of π holds for all minimal hypertorus graphs Γ such that |V` | < k and takethe minimal hypertorus graph Γ such that |V` | = k. From Lemma 17.4, we can take p ∈ V`

such taht Γ ′ = Γ −N(p) is connected and L−L∩N(p) is connected for all codimension twohypertorus subgraphs L. Now we have, for the edge (or leg) e in Γ ′, there is a codimensiontwo hypertorus graph L ′ of Γ ′ from Proposition 16.7 such taht e is a normal edge (leg) ofL ′. Moreover for this e we can take codimension two hypertorus subgraph L of Γ such tahte is a normal edge (leg) of L. Then we see L ′ = L − L ∩N(p). Because of our assumption,we can take hyperfacets H and H such that ∂H = L. We also have H ′ = H − H ∩ N(p)is a hyperfacet of Γ ′ such that ∂H ′ = L ′. Hence Γ ′ is a minimal hypertorus graph and itsatsfies our assumptions. Moreover we have |V`

′| = k − 1, so we have the following map

π ′ ◦ r is surjective from the assumption of the induction:

Z[x, H1, · · · , Hm]ı−→ HT (Γ, α)

r ↓ ↓ r ′

Z[x, H ′1, · · · , H ′

l]ı ′−→ HT (Γ

′, α|E` ′ ),

where r(H) = H ∩ Γ ′, r(x) = x and r ′(f) = f|` ′ . Hence we have r ′ ◦ π = π ′ ◦ r is surjective.So all f|` ′ are denoted by Z[x,H1, · · · , Hm] as identified H ′(= H ∩ Γ ′) and H.

Because we see g = f − f|` ′ is in H∗T (Γ, α) and g(q) = 0 for all q 6= p, the following

equation holds from the definition of H∗T (Γ, α) and the definition of the hyperfacet:

g(p) = k∏

pq∈E`p

α(pq) = k∏

pq∈E`p

τHq(p),

where Hq is the hyperfacet whose normal edge on p is pq and some element k ∈ H∗T (pt).

Moreover we have the vertices of X = ∩pq∈E`pHq is only one point p that is

{p} = VX

from the assumption that H∩H ′ = ∅ or coonected for all hyper facets H and H ′. Thereforewe see

g = k∏

pq∈E`p

Hq.

Hence we have f = f|` ′ + g ∈ Im(π). ¤83

Hence we have the following theorem from Lemma 17.2 and 17.5.

THEOREM 17.2. Let Γ be a minimal hypertorus graph. If it holds H1 ∩ H2 = ∅ or connectedfor every hyperfacet H1 and H2 in Γ and there is a unique pair {H, H} such that H∩ H = L for allcodimension two hypertorus subgraphs L, then we have Z[Γ, θ] ' H∗

T (Γ, α).

17.3. Proof of the main theorem.In this section we will prove the main theorem. To prove it, we will use an inductive

argument for |V` | and the Mayer-Vietoris analogue.First of all we can assume the statement of Main Theorem 2 holds for all hypertorus

graphs Γ such that |V` | < k−1 because we have already known Main Theorem 2 holds for|V` | = 1 by Theorem 17.1. We also have already known the statement of Main Theorem2 holds for the minimal hypertorus graph by Theorem 17.2. So there is the codimensiontwo hypertorus subgraph L ⊂ Γ which has the unique hyperfacet H and H such thatH ∩ H = L and N(H), N(H) 6= Γ . Put these neighborhood N(H) = Γ1, N(H) = Γ2 andΓ1 ∩ Γ2 = N(L) = Γ3. Then the graph Γi = (V`i, E`i) is the hypertorus graph which hasa restricted connection θ|`i and a restricted axial function α|`i of the hypertorus graph(Γ, α, θ) and all i = 1, 2, 3. Since we assume the condition H ∩ G = ∅ or connected forall hyperfacets H and G of (Γ, α, θ), H ∩ Γi is connected for every hyperfacet H of (Γ, α, θ).Hence all hyperfacets of Γi are inherited from hyperfacets of (Γ, α, θ), that is the set of allhyperfacets of Γi is a set Hi = {Γi ∩ H | H ∈ H} for i = 1, 2, 3. So we also have thesehypertorus graph satisfies conditions as follows:

(1) There is a unique hyperfacet H and its opposite side H for every codimension twohyperfacet L in (Γi, α|`i , θ|`i) such that ∂H = L.

(2) For all hyperfacets H and G, these intersection H ∩G = ∅ or connected.From the assumption of the induction, we also have Ψi : Z[Γi, θ|`i] → H∗

T (Γi, α|`i) is iso-morphic for each i = 1, 2, 3.

Put the homomorphism ρ1 : Z[Γ, θ] → Z[Γ1, θ|`1 ]⊕ Z[Γ2, θ|`2 ] such that

ρ1(H) = Γ1 ∩H⊕ Γ2 ∩H,

ρ1(x) = x⊕ x,

and ρ2 : Z[Γ1, θ|`1]⊕ Z[Γ2, θ|`2] → Z[Γ3, θ|`3] such that

ρ2(H1 ⊕H2) = Γ3 ∩H1 − Γ3 ∩H2,

ρ2(x⊕H2) = x − Γ3 ∩H2,

ρ2(H1 ⊕ x) = Γ3 ∩H1 − x,

where H (resp. Hi) is a hyperfacet of Γ (resp. Γi) and assume if Γi∩H = ∅ then Γi∩H = 0 inZ[Γi, θ|`i ]. Because all hyperfacets in Γi are inherited from Γ , these maps are well-defined.

Then ρ1 is injective because we have Ker(p1 ◦ ρ1) ∩ Ker(p2 ◦ ρ1) = {0} = Ker(ρ1) bythe definition of the assumption if Γi ∩H = ∅ then Γi ∩H = 0 in Z[Γi, θ|`i ] and Γ1 ∪ Γ2 = Γ ,where pi = Z[Γ1, θ|`1 ]⊕ Z[Γ2, θ|`2] → Z[Γi, θ|`i ].

84

Because all hyperfacets of Γ3 are inherited from Γ1, we can get all generators G ofZ[Γ3, θ|`3 ] by ρ2(H ⊕ 0) = Γ3 ∩ H for some generator H ∈ Z[Γ1, θ|`1]. So we see ρ2 issurjective.

Moreover we have the following lemma.

LEMMA 17.6. The following sequence is exact:

{0} −→ Z[Γ, θ]1−→ Z[Γ1, θ|`1 ]⊕ Z[Γ2, θ|`2]

2−→ Z[Γ3, θ|`3] −→ {0}.

PROOF. We may only show Im(ρ1) = Ker(ρ2). First we can get Im(ρ1) ⊂ Ker(ρ2) fromthe following equation:

ρ2 ◦ ρ1(X)

= ρ2(X ∩ Γ1 ⊕ X ∩ Γ2)

= X ∩ Γ3 − X ∩ Γ3

= 0.

Next we assume generators H1 ∈ Z[Γ1, θ|`1] and H2 ∈ Z[Γ2, θ|`2] satisfy Γ3 ∩ H1 − Γ3 ∩H2 = 0. Then this means the hyperfacet H1 of Γ1 coinsides with the hyperfacet H2 ofΓ2 on the hypertorus graph Γ3. So H1 ∪ H2 = H is a hyperfacet of Γ . Hence we haveIm(ρ1) ⊃ Ker(ρ2). ¤

Next we consider the equivariant graph cohomologies H∗T (Γ, α) and H∗

T (Γi, α|`i) (i =1, 2, 3).

Put the homomorphism ρ ′1 : H∗T (Γ, α) → H∗

T (Γ1, α|`1)⊕H∗T (Γ2, α|`2) such that

ρ ′1(f) = f|`1 ⊕ f|`2

and ρ ′2 : H∗T (Γ1, α|`1)⊕H∗

T (Γ2, α|`2) → H∗T (Γ3, α|`3) such that

ρ ′2(g⊕ h) = g|`3 − h|`3 .

Now ρ ′1 is injective because we see if ρ ′1(f) = 0 then f(p) = 0 for all p ∈ V`1 ∪ V`2 = V` .Moreover we have the following lemma.

LEMMA 17.7. The following sequence is exact:

{0} −→ H∗T (Γ, α)

′1−→ H∗T (Γ1, α|`1)⊕H∗

T (Γ2, α|`2) ′2−→ H∗

T (Γ3, α|`3).

PROOF. First we have ρ ′2 ◦ ρ ′1(f) = f|`3 − f|`3 = 0, so Im(ρ ′1) ⊂ Ker(ρ ′2) holds. Nextwe take g ⊕ h ∈ Ker(ρ ′2), then g|`3 = h|`3 . Hence the following map f : V` → H∗

T (pt) iswell-defined and in H∗

T (Γ, α):

f(p) = g(p) if p ∈ V`1 ,

f(q) = h(q) if q ∈ V`2 .

So we have Im(ρ ′1) ⊃ Ker(ρ ′2). ¤

85

LEMMA 17.8. The following diagram is commutative:

{0} −→ Z[Γ, θ]1−→ Z[Γ1, θ|`1]⊕ Z[N(Γ2), θ|`2]

2−→ Z[Γ3, θ|`3 ]↓ Ψ ↓ Ψ1 ⊕ Ψ2 ↓ Ψ3 ↓{0} −→ H∗

T (Γ, α) ′1−→ H∗

T (Γ1, α|`1)⊕H∗T (Γ1), α|`2)

′2−→ H∗T (Γ3, α|`3)

PROOF. Now we see ρ ′1(τH) = τH∩`1⊕τH∩`2 , ρ′1(x) = x⊕x by the definition of ρ ′1. Hence

we see the left square is commute from the definitions of ρ1, Ψ, Ψ1 and Ψ2. Similarly wehave the right square is commute by definitions of ρ2 and ρ ′2. ¤

From the assumption of the induction, we have Ψ1⊕Ψ2 and Ψ3 are isomorphic. Hencewe have Ψ is isomorphic from the above Lemma 17.6 through 17.8 and the five lemma.Therefore we have the Main Theorem 2.

Finally we exhibit two examples which does not satisfies two assumptions of MainTheorem 2 that is

(1) There is a unique hyperfacet H and its opposite side H for every codimension twohyperfacet L in (Γ, α, θ) such that ∂H = L.

(2) For all hyperfacets H and G, these intersection H ∩G = ∅ or connected.

FIGURE 17.1. The figure which does not satisfy the assumption 1.

86

β α

−α+x −β+x

α

−β+x−α+x

β

FIGURE 17.2. The figure which does not satisfy the assumption 2.

On the above two cases Main Theoreom 2 deos not hold, that is

Z[Γ, θ] 6' H∗T (Γ, α).

87

Bibliography

[Ada69] J. F. Adams: Lectures on Lie Groups, Benjamin, 1969.[Aso81] T. Asoh: Compact transformation groups on Z2-cohomology spheres with orbits of codimension 1, Hi-

rosima Math. J., 11 (1981), 571–616.[BD00] R. Bielowski, A. Dancer: The geometry and topology of toric hyperKaler, Com. Anal. Geom., Vol. 8, No.

4 (2000), 727–759.[BH58] A. Borel, F. Hirzebruch: Characteristic classes and homogeneous spaces I, Amer. J. Math., Vol. 80, No. 2

(1958), 458–538.[Bre72] G. E. Bredon: Introduction to compact transformation groups, Academic Press, 1972.[BJ82] TH. Brocker, K. Janich: Introduction to differetial topology, Cambridge Univ. Press, 1982.[BP02] V. M. Buchstaber, T. E. Panov: Torus actions and their applications in topology and combinatorics,

Amer. Math. Soc., 2002.[CS74] T. Chang, T. Skjelbret: The topological Schur lemma and related result, Ann. Math., 100 (1974), 307–321.[DJ91] M. Davis, T. Januszkiewicz: Convex polytopes, Coxeter orbifolds and torus action, Duke. Math. J., 62

(1991), no. 2, 417–451.[GKM98] M. Goresky, R. Kottwitz, R. Macpherson: Equivariant cohomology, Koszul duality, and the localization

theorem, Invent. Math., 131 (1998), 25–83.[GZ01] V. Guillemin, C. Zara: 1-skeleta, Betti numbers, and equivariant cohomology, Duke Math. J., 107 (2001),

283–349.[HH04] M. Harada, T. S. Holm: The equivariant cohomology of hypertoric varieties and their real loci, arXiv:

math. DG/0405422.[HM03] A. Hattori, M. Masuda: Theory of Multi-fans, Osaka J. Math., 40 (2003), 1–68.[HH65] W. C. Hsiang, W. Y. Hsiang: Classification of differentiable actions on Sn, Rn and Dn with Sk as the

principal orbit type, Ann. Math., 82 (1965), 421–433.[HKLR87] N. Hitchin, A. Karlhede, U. Lindstrom, M. Rocek: Hyperkahler metrics and supersymmetry, Com-

mun. Math. Phys. 108 (1987), 535–589.[Iwa78] K. Iwata: Compact transformation groups on cohomology quaternion projective spaces with codimension

one orbits, Osaka J. Math., 15-3 (1978), 475–508.[Iwa81] K. Iwata: Compact transformation groups on raional cohomology Cayley projective planes, Tohku Math.

J. (2), 33 (1981), no. 4, 429–442.[Kaw91] K. Kawakubo: The theory of transformation groups, Oxford Univ. Press, London, 1991.[Kon00] H. Konno: Cohomology rings of toric hyperKahler manifolds, Int. J. of Math., 11, no. 8 (2000), 1001–

1026.[Kon03] H. Konno: Variation of toric hyperKahler manifolds, Int. J. of Math., 14 (2003), 289–311.[Kol02] A. Kollross: A Classification of hyperpolar and cohomogeneity one actions, Trans. Amer. Math. Soc., 354

(2002), no. 2, 571–612.[Kur1] S. Kuroki: On the construction of smooth SL(m,H)×SL(n,H)-actions on S4(m+n)−1, Bull. of Yamagata

Univ., Nat. Sci., Vol.15, No.3 Feb. 2003, 49–59.[Kur2] S. Kuroki: Classification of compact transformation groups on complex quadrics with codimension one orbits,

submitted.

89

[Kur3] S. Kuroki: On SL(3,R)-action on 4-sphere, to appear in the Journal of Fundamental and AppliedMathematics.

[Kur4] S. Kuroki: Equivariant graph cohomology of hypertorus graph and (n + 1)-dimensional torus action on4n-dimensional manifold, preprint.

[MMP05] H. Maeda, M. Masuda, T. Panov: Torus graphs and simplicial posets, arXiv: math. AT/0511582.[Mas99] M. Masuda: Unitary toric manifolds, Multi-fans and Equivariant index, Tohku Math. J., 51 (1999),

237–265.[MP03] M. Masuda, T. Panov: On the cohomology of torus manifolds, arXiv: math. AT/0306100.[MS74] J. W. Milnor, J. D. Stasheff: Characteristic classes, Princeton Univ. Press, 1974.[MS43] D. Montgomery, H. Samelson: Transformation groups on spheres, Ann. of Math., 44 (1943), 454–470.[Nak84] A. Nakanishi: SO(n), SU(n), Sp(n)-homology spheres with codimension two principal orbits, Tokyo. J.

Math., Vol. 7, No. 2, 1984.[Har90] F. Reese Harvey: Spinors and Calibrations, Academic Press, 1990.[TM] H. Toda, M. Mimura: Topology of Lie groups (Japanese), Kinokuniyasyoten, 1978, 1979.[Uch77] F. Uchida: Classification of compact transformation groups on cohomology complex projective spaces with

codimension one orbits, Japan J. Math., Vol. 3, No. 1 (1977), 141–189.[Uch78] F. Uchida: Compact transformation groups on complex projective spaces with codimension two principal

orbits, Osaka J. Math., 15(1978), 137–150.[Wan49] H. C. Wang: Homogeneous spaces with non-vanishing Euler characteristics, Ann. of Math., 50 (1949),

925–953.[Wan60] H. C. Wang: Compact transformation groups of Sn with an (n − 1)-dimensional orbit, Amer. J. Math.,

82 (1960), 698–748.[Yok73] I. Yokota: Groups and Representations (Japanese), Shokabou, 1973.

90